U.S. patent application number 10/287173 was filed with the patent office on 2003-11-20 for methods and systems for the identification of components of mammalian biochemical networks as targets for therapeutic agents.
Invention is credited to Hill, Colin, Khalil, Iya.
Application Number | 20030215786 10/287173 |
Document ID | / |
Family ID | 29424453 |
Filed Date | 2003-11-20 |
United States Patent
Application |
20030215786 |
Kind Code |
A1 |
Hill, Colin ; et
al. |
November 20, 2003 |
Methods and systems for the identification of components of
mammalian biochemical networks as targets for therapeutic
agents
Abstract
Systems and methods for modeling the interactions of the several
genes, proteins and other components of a cell, employing
mathematical techniques to represent the interrelationships between
the cell components and the manipulation of the dynamics of the
cell to determine which components of a cell may be targets for
interaction with therapeutic agents. A first such method is based
on a cell simulation approach in which a cellular biochemical
network intrinsic to a phenotype of the cell is simulated by
specifying its components and their interrelationships. The various
interrelationships are represented with one or more mathematical
equations which are solved to simulate a first state of the cell.
The simulated network is then perturbed by deleting one or more
components, changing the concentration of one or more components,
or modifying one or more mathematical equations representing the
interrelationships between one or more of the components. The
equations representing the perturbed network are solved to simulate
a second state of the cell which is compared to the first state to
identify the effect of the perturbation on the state of the
network, thereby identifying one or more components as targets. A
second method for identifying components of a cell as targets for
interaction with therapeutic agents is based upon an analytical
approach, in which a stable phenotype of a cell is specified and
correlated to the state of the cell and the role of that cellular
state to its operation. A cellular biochemical network believed to
be intrinsic to that phenotype is then specified by identifying its
components and their interrelationships and representing those
interrelationships in one or more mathematical equations. The
network is then perturbed and the equations representing the
perturbed network are solved to determine whether the perturbation
is likely to cause the transition of the cell from one phenotype to
another, thereby identifying one or more components as targets.
Inventors: |
Hill, Colin; (Ithaca,
NY) ; Khalil, Iya; (Ithaca, NY) |
Correspondence
Address: |
KRAMER LEVIN NAFTALIS & FRANKEL LLP
INTELLECTUAL PROPERTY DEPARTMENT
919 THIRD AVENUE
NEW YORK
NY
10022
US
|
Family ID: |
29424453 |
Appl. No.: |
10/287173 |
Filed: |
November 4, 2002 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
60335999 |
Nov 2, 2001 |
|
|
|
60406764 |
Aug 29, 2002 |
|
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Current U.S.
Class: |
435/4 ;
703/11 |
Current CPC
Class: |
G16B 5/10 20190201; G01N
2800/52 20130101; Y10S 707/99943 20130101; G16B 5/20 20190201; G16B
5/00 20190201; G16B 5/30 20190201 |
Class at
Publication: |
435/4 ;
703/11 |
International
Class: |
C12Q 001/00; G06G
007/48; G06G 007/58 |
Claims
What is claimed:
1. A method for identifying one or more components of a cell as
putative targets for interaction with one or more agents,
comprising the steps of: (a) specifying a biochemical network
believed to be intrinsic to a phenotype of said cell; (b)
simulating said network by (i) specifying the components of said
network, and (ii) representing interrelationships between said
components in one or more mathematical equations; (c) solving the
mathematical equations to simulate a first state of the cell; (d)
perturbing the simulated network by deleting one or more components
thereof, changing the concentration of one or more components
thereof or modifying one or more mathematical equations
representing interrelationships between one or more of said
components; (e) solving the equations representing the perturbed
network to simulate a second state of the cell; and (f) comparing
said first and second simulated states of the network to identify
the effect of said perturbation on the state of the network, and
thereby identifying one or more components for interaction with one
or more agents.
2. A method as recited in claim 1 wherein said mathematical
equations are solved using stochastic or differential
equations.
3. A method as recited in claim 1 wherein the concentrations of one
or more of the several proteins and genes in the biochemical
network are selectively perturbed to identify which ones of said
proteins or genes cause a change in the time course of the
concentration of a gene or protein implicated in a disease state of
said cell.
4. A method as recited in claim 3 wherein a series of perturbations
are made, each of said perturbations changing the concentration of
a protein or gene in said network to a perturbed value, to
determine whether that protein or gene is implicated in causing a
change in the time course of the concentration of a gene or protein
implicated in a disease state of said cell.
5. A method as recited in claim 4 wherein the concentration of each
of said proteins and genes is reduced to zero in each respective
perturbation.
6. A method as recited in claim 1 wherein the concentrations of one
or more of the components of the said biochemical network are
optimized by determining the minima or multiple minima of said
concentrations.
7. A method as recited in claim 1 wherein the concentrations and or
the parameters of one or more of the several proteins and genes in
the biochemical network are systematically perturbed to identify
which ones of said proteins or genes cause a change in the time
course of the concentration of a gene or protein implicated in a
disease state of said cell.
8. A method for identifying one or more components of a cell as
putative targets for interaction with one or more agents,
comprising the steps of: (a) specifying a stable phenotype of a
cell; (b) correlating said phenotype to the state of the cell; (c)
specifying a cellular biochemical network believed to be intrinsic
to said phenotype; (d) characterizing said network by (i)
specifying the components thereof, and (ii) specifying
interrelationships between said components and representing said
interrelationships in one or more mathematical equations; (e)
perturbing the characterized network by deleting one or more
components thereof, changing the concentration of one or more
components thereof or modifying one or more mathematical equations
representing interrelationships between one or more of said
components; and (f) solving the equations representing the
perturbed network to determine whether said perturbation is likely
to cause the transition of said cell from one phenotype to another,
and thereby identifying one or more components for interaction with
one or more agents.
9. A method as recited in claim 8 wherein the stable attractors
include at least one of an equilibrium state characterized by
steady state values, a periodically changing state characterized by
periodically changing values, and a chaotically changing state
having a peculiar signature.
10. A method as recited in claim 8 wherein the concentrations of
one or more of the several proteins and genes in the biochemical
network are selectively perturbed to identify which ones of said
proteins or genes are implicated in causing an attractor of the
biochemical network to become unstable.
11. A method as recited in claim 8 comprising carrying out a series
of perturbations, each of said perturbations changing the
concentration of a protein or gene in said network to a perturbed
value to determine whether that protein or gene is implicated in
causing a change in the time course of the concentration of a gene
or protein implicated in a disease state of said cell.
12. A method as recited in claim 8 wherein the concentration of
each of said proteins and genes is reduced to zero in each
respective perturbation.
13. A method as recited in claim 8 wherein a bifurcation analysis
is performed using eigen values of a Jacobian matrix based upon
said equations to characterize the stability of one or more
attractors.
14. A method as recited in claim 8 wherein the step of
characterizing said network includes at least one of: (iii)
identifying new and missing links and components in the network;
(iv) constraining parameter values in the network; and (v)
determining parameter values in the network.
15. A method for identifying one or more components of a cell as
putative targets for interaction with one or more agents,
comprising the steps of: (a) specifying a biochemical network
believed to be intrinsic to a phenotype of said cell; (b) inferring
new links and components in the network using experimental data;
(c) simulating said network by (i) specifying the components of
said network, and (ii) specifying interrelationships between said
components and representing said interrelationships in one or more
mathematical equations; (d) inferring new and missing links and
components in the network; (e) constraining and or determining
parameter values in the network by (i) sampling a set of networks
and parameter values, (ii) simulating the said networks as
described in (c), and (iii) determining the network and parameter
values that optimally fits a given set or sets of experimental
data; (f) solving those equations representing the network to
simulate a first state of the cell; (g) perturbing the simulated
network by deleting one or more components thereof, changing the
concentration of one or more components thereof or modifying one or
more mathematical equations representing interrelationships between
one or more of said components; (h) solving the equations
representing the perturbed network to simulate a second state of
the cell; and (i) comparing said first and second simulated states
of the network to identify the effect of said perturbation on the
state of the network, and thereby identifying one or more
components for interaction with one or more agents.
16. A method as recited in claim 15 wherein the experimental data
includes at least one of a DNA sequence, protein sequence,
microarray data, expression data, time course expression data, and
a protein structure.
17. A method as recited in claim 1 including the steps of storing
said mathematical formulae in computer memory, storing algorithms
in computer memory for solving said mathematical formulae, said
solving step or steps each comprising retrieving said algorithms
and applying them to solve said formulae.
18. A method as recited in claim 17 in which said perturbing step
includes storing in computer memory a plurality of values for use
in said perturbing step, and using an algorithm to apply said
values separately or in combination with one another to
automatically change the perturbations in accordance with a
predetermined sequence.
19. A method as recited in claim 8 including the steps of storing
said mathematical formulae in computer memory, storing algorithms
in computer memory for solving said mathematical formulae, said
solving step or steps each comprising retrieving said algorithms
and applying them to solve said formulae.
20. A method as recited in claim 19 in which said perturbing step
includes storing in computer memory a plurality of values for use
in said perturbing step and using an algorithm to apply said values
separately or in combination with one another to automatically
change the perturbations in accordance with a predetermined
sequence.
21. A method as recited in claim 15 including the steps of storing
said mathematical formulae in computer memory, storing algorithms
in computer memory for solving said mathematical formulae, said
solving step or steps each comprising retrieving said algorithms
and applying them to solve said formulae.
22. A method as recited in claim 21 in which said perturbing step
includes storing in computer memory a plurality of values for use
in said perturbing step and using an algorithm to apply said values
separately or in combination with one another to automatically
change the perturbations in accordance with a predetermined
sequence.
23. A method as recited in claim 1 wherein experiments are
conducted to confirm the identified component as a target.
24. A method as recited in claim 8 wherein experiments are
conducted to confirm the identified component as a target.
25. A method for creating an optimized mathematical simulation of a
biochemical network of a cell comprising: (a) specifying a
biochemical network of a cell; (b) simulating said network by (i)
specifying the components of said network, and (ii) representing
interrelationships between said components in one or more
mathematical equations and setting the quantitative parameters of
said components; and (c) optimizing said simulated biochemical
network by determining and constraining the parameter values set
therein.
26. A method for creating an optimized mathematical simulation of a
biochemical network of a cell as recited in claim 25 wherein
optimization algorithms are used to constrain the parameter values
to fit the measured data.
27. A method as recited in claim 25 comprising the further steps of
(d) fitting the parameter values to said data and assessing how
good the fit is; and (e) performing an error analysis to determine
it there are other parameter values or the populatoin of parameter
values which fit the data but yield a different prediction and
identifying that prediction. (f) experimentally verifying
predictions from the model in order to validate a single prediction
or disceren between various predictions or hypotheses and/or using
the experimentally derived results to iteratively refine the
model.
28. A method as recited in claim 27 wherein experiments are
conducted to validate that prediction.
29. A method as recited in claim 26 including the steps of storing
said mathematical formulae in computer memory, storing said
optimization algorithms in computer memory, storing in computer
memory values corresponding to said quantitative parameters, and
applying said algorithms to said parameters to optimize said
simulated biochemical network.
30. A method for identifying one or more components of a cell as
putative targets for interaction with one or more agents,
comprising the steps of: (a) specifying a biochemical network of a
cell; (b) simulating said network by (i) specifying the components
of said network, and (ii) representing interrelationships between
said components in one or more mathematical equations and setting
the quantitative parameters of said components; and (c) optimizing
said simulated biochemical network by determining and constraining
the values of the parameters of said components; and (d) solving
the mathematical equations to simulate a state of said cell.
31. A method as recited in claim 30 including the steps of storing
said mathematical formulae in computer memory, storing algorithms
in computer memory for solving said mathematical formulae, said
solving step or steps each comprising retrieving said algorithms
and applying them to solve said formulae.
32. A method as recited in claim 31 including the step of storing
optimization algorithms in computer memory, storing in computer
memory values corresponding to said quantitative parameters, and
applying said algorithms to said parameters to optimize said
simulated biochemical network.
33. A method of predicting the physiological state of a cell
comprising the steps of: (a) specifying a biochemical network of a
cell; (b) simulating said network by (i) specifying the components
of said network, and (ii) representing interrelationships between
said components in one or more mathematical equations and setting
the quantitative parameters of said components; (c) optimizing said
first simulated biochemical network by determining and constraining
the values of the parameters of said components; and (d)
determining the state of said cell by solving the mathematical
equations and thereby simulating the physiological state of said
cell.
34. A method as recited in claim 33 wherein said cell is a cancer
cell.
35. A method as recited in claim 33 in which said perturbing step
includes storing in computer memory a plurality of values for use
in said perturbing step and using an algorithm to apply said values
separately or in combination with one another to automatically
change the perturbations in accordance with a predetermined
sequence.
36. A method as recited in claim 35 including the step of storing
optimization algorithms in computer memory, storing in computer
memory values corresponding to said quantitative parameters, and
applying said algorithms to said parameters to optimize said
simulated biochemical network.
37. A method of predicting an altered physiological state of a cell
comprising the steps of: (a) specifying a biochemical network of a
cell; (b) simulating said network by (i) specifying the components
of said network, and (ii) representing interrelationships between
said components in one or more mathematical equations and setting
the quantitative parameters of said components; (c) optimizing said
first simulated biochemical network by determining and constraining
the values of the parameters of said components; (d) perturbing the
optimized simulated network by adding or deleting one or more
components thereof, changing the concentration of one or more
components thereof or modifying one or more mathematical equations
representing interrelationships between one or more of said
components; (e) solving the equations representing the perturbed
network to simulate a second state of the cell; and (f) comparing
said first and second simulated states of the network to identify
the effect of said perturbation on the state of the network.
38. A method as recited in claim 37 wherein the simulated network
is systematically perturbed.
39. A method as recited in claim 37 wherein the simulated network
is systematically perturbed by deleting two or more components.
40. A method as recited in claim 37 wherein the physiological state
is proliferation.
41. A method as recited in claim 37 wherein said physiological
state is G1-S and wherein Cyclin E-CDK2 is used as the marker for
said determination.
42. A method as recited in claim 37 wherein said physiological
state is G2-M and wherein Cyclin B-CDK1 is used as the marker for
said determination.
43. A method as recited in claim 37 wherein said physiological
state is S phase arrest and wherein Cyclin A-CDK2 is used as the
marker for said determination.
44. A method as recited in claim 37 wherein said physiological
state is apoptosis and wherein caspase 3 and cleaved PARP are the
markers of said state.
45. A method as recited in claim 37 including the steps of storing
said mathematical formulae in computer memory, storing algorithms
in computer memory for solving said mathematical formulae, said
solving step or steps each comprising retrieving said algorithms
and applying them to solve said formulae including the step of
storing optimization algorithms in computer memory, storing in
computer memory values corresponding to said quantitative
parameters, and applying said algorithms to said parameters to
optimize said simulated biochemical network.
46. A method as recited in claim 45 in which said perturbing step
includes storing in computer memory a plurality of values for use
in said perturbing step and using an algorithm to apply said values
separately or in combination with one another to automatically
change the perturbations in accordance with a predetermined
sequence.
47. A method of simulating the physiological state of a cancer cell
comprising the steps of: (a) specifying a biochemical network of a
cell; (b) simulating said network by (i) specifying the components
of said network, and (ii) representing interrelationships between
said components in one or more mathematical equations and setting
the quantitative parameters of said components; and (c) solving the
mathematical equations to simulate a first state of the cell.
48. A method as recited in claim 47 wherein the physiological state
is manifested by Erk a high level of proliferative signals.
49. A method as recited in claim 48 wherein said signal is Erk.
50. A method as recited in claim 47 wherein said physiological
state is manifested by a high level of pro-apoptotic proteins.
51. A method as recited in claim 50 wherein said pro-apoptotic
protein is Bcl2.
52. A method as recited in claim 47 wherein after simulating the
first state of the cell the method further comprises: (d)
perturbing the simulated network by deleting one or more components
thereof, changing the concentration of one or more components
thereof or modifying one or more mathematical equations
representing interrelationships between one or more of said
components; (e) solving the equations representing the perturbed
network to simulate a second physiological state of the cell; and
(f) comparing said first and second simulated states of the network
to identify the effect of said perturbation on the state of the
network.
53. A method as recited in claim 52 wherein the second simulated
state of the network is analyzed to determine whether the cells
have gone through G1-S arrest, G2-M arrest, S phase arrest and/or
apoptosis.
54. A method as recited in claim 52 wherein said method is used to
predict the sensitivity of said cell to a particular state.
55. A method as recited in claim 53 wherein said state is
apoptosis.
56. A method is recited in claim 52 wherein in step (e) two or more
components are perturbed.
57. A method as recited in claim 47 including the steps of storing
said mathematical formulae in computer memory, storing algorithms
in computer memory for solving said mathematical formulae, said
solving step or steps each comprising retrieving said algorithms
and applying them to solve said formulae.
58. A method as recited in claim 57 in which said perturbing step
includes storing in computer memory a plurality of values for use
in said perturbing step and using an algorithm to apply said values
separately or in combination with one another to automatically
change the perturbations in accordance with a predetermined
sequence.
59. A method for testing a substance for possible use as a
therapeutic by simulating its effect on the physiological state of
a cell, comprising the steps of: (a) specifying a biochemical
network of a cell; (b) simulating said network by (i) specifying
the components of said network, and (ii) representing
interrelationships between said components in one or more
mathematical equations and setting the quantitative parameters of
said components; and (c) solving the mathematical equations to
simulate a first physiological state of the cell; (d) modifying the
simulated network created in step (b) by representing the
interrelationships between said chosen substance and other cell
components in mathematical equations and setting forth the
quantitative parameters of said components; (e) solving the
mathematical equations of said modified simulated network.
60. A method as recited in claim 59 further comprising perturbing
the modified simulated network by deleting one or more components
thereof, changing the concentration of one or more components
thereof or modifying one or more mathematical equations
representing interrelationships between one or more of said
components.
61. A method as recited in claim 59 wherein said substance is
exogenous to said cell.
62. A method as recited in claim 59 in which said perturbing step
includes storing in computer memory a plurality of values for use
in said perturbing step and using an algorithm to apply said values
separately or in combination with one another to automatically
change the perturbations in accordance with a predetermined
sequence.
63. A method as recited in claim 60 in which said perturbing step
includes storing in computer memory a plurality of values for use
in said perturbing step and using an algorithm to apply said values
separately or in combination with one another to automatically
change the perturbations in accordance with a predetermined
sequence.
64. A method as recited in claim 59 wherein experiments are
conducted to confirm the therapeutic value of a substance
identified by the method.
65. An iterative method of from 2-n steps for simulating the
physiological state of a cell under iteratively modified conditions
comprising the steps of (a) specifying a biochemical network of a
cell; (b) simulating said network by (i) specifying the components
of said network, and (ii) representing interrelationships between
said components in one or more mathematical equations and setting
the quantitative parameters of said components; and (c) solving the
mathematical equations to simulate a first state of the cell; (d)
perturbing the simulated network by adding or deleting one or more
components thereof, changing the concentration of one or more
components thereof or modifying one or more mathematical equations
representing interrelationships between one or more of said
components; (e) solving the equations representing the perturbed
network to simulate a second physiological state of the cell; (f)
comparing said first and second simulated states of the network to
identify the effect of said perturbation on the state of the
network; and (g) repeating steps (d)-(f) from one to n times to
create further modified simulated networks.
Description
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] This application claims the benefit of the following United
States Provisional Patent Applications: "Methods and Systems for
the Identification of Components of mamallian Cells as Targets for
Therapeutic Agents", U.S. Provisional Patent Application Serial No.
60/335,999, filed on Nov. 2, 2001; and "Systems and Methods For
Inferring Biological Networks", Vipul Periwal, Inventor, U.S.
Provisional Patent Application Serial No. 60/406,764, filed on Aug.
29, 2002.
[0002] This application also claims priority to the following
United States Patent Application: "Scale-Free Network Inference
Methods", Jeff Fox, Colin Hill andVipul Periwal, Inventors, U.S.
application Ser. No. ______, filed on Nov. 1, 2002 (serial number
to be added by amendment when available).
FIELD OF THE INVENTION
[0003] The present invention relates to drug discovery. More
particularly, the invention relates to in silico methods for
identifying one or more components of a cell as a target for
interaction with one or more therapeutic agents. Even more
specifically, the invention relates to methods for the simulation
or analysis of the dynamic interrelationships of genes and proteins
with one another and to the use of those methods to identify one or
more cellular components as putative targets for a therapeutic
agent.
BACKGROUND OF THE INVENTION
[0004] Drug Discovery
[0005] Several methods have been employed to find therapeutic
compounds useful in the treatment of disease states. Typically,
these methods involve empirical studies of organisms or cells and
in some cases the components of cells to identify active
therapeutic compounds which themselves, or in modified form, may
have a beneficial effect on the organism or cell.
[0006] Screening methods have been used to find compounds that have
a sought after-effect on a cell, i.e. the up regulation or down
regulation of a gene. Screening assays are used to identify
compounds and those which are identified may be used in further
drug development activity. Using such methods, for example,
antibodies that bind to receptors on animal tumor cells may be
assayed and identified. In further drug development efforts, these
antibodies or their epitopes can be analyzed and their therapeutic
activity enhanced by methods known in the art.
[0007] A difficulty with such methods is that they are basically
brute-force empirical methods that reveal little or nothing about
the particular phenomena which take place within the cell when it
is contacted with the compound identified in the screen. The actual
cellular dynamics may not be understood and this may lead to
development of candidate drugs deleteriously, which affect other
components in the cell and cause undesirable side effects. This
brute-force screening method is also limited by the speed at which
assays can be conducted.
[0008] Another empirical approach used in drug development is that
of screening compounds against a particular component of a cell
which has been identified as being involved in a disease condition.
Assays are conducted to determine the binding effect, chemical
interaction or other modification of certain molecules within the
cell such as genes or proteins. While the art has developed
powerful, high throughput screening techniques by which tens of
thousands of compounds are routinely screened for their interactive
effect with one or more targets, such methodologies are still
inherently empirical and leave the researcher with no fundamental
information about the mechanisms of interaction of a compound
identified by such methods. Thus the compound so identified may
have detrimental interactions with one or more other components of
a cell and may cause more harm than good. In order to determine
whether the so identified compound may ultimately be useful as a
therapeutic, it must be tested using in vitro studies on cells
containing the particular gene or protein with which it interacts,
or in vitro animal studies to determine both its beneficial and
possible detrimental effects. These additional tests are extremely
time-consuming and expensive.
[0009] Recently, investigators have sought to make the drug
discovery process more rational by exploring the effects of a drug
under development on various modifications of cells and cellular
components. Friend et al., U.S. Pat. No. 6,165,709 describes
methods for identifying the cellular targets of a drug by comparing
(i) the effects of the drug on a wild-type cell, (ii) the effects
on a wild-type cell of modifications to a putative target of the
drug, and (iii) the effects of the drug on a wild-type cell which
has had the putative target modified. The effect of the drug on the
cell can be determined by measuring various aspects of the cell
state, including gene expression, concentration of proteins, etc.
While the methods described are improved over the brute-force
empirical methods described above, multiple "wet" experiments must
be conducted in order to determine that a gene or protein component
of a cell is in fact a target of a drug and then to determine the
effects of the drug on that component, on a modified component and
on a wild-type cell.
OBJECTS OF THE INVENTION
[0010] It is an object of the invention to expedite the drug
discovery process and to avoid costs and delays of present
drug-screening methods.
[0011] It is a further and related object of the invention to
reduce labor and equipment costs of empirical drug discovery
processes.
[0012] It is still a further object of the invention to reduce or
avoid the need for setting up expensive in vitro and in vivo
experiments to determine the efficacy, toxicity and side effects of
drug candidates.
[0013] It is still a further object of the invention to determine
the effects of drug candidates on the cell as a whole or the least
upon a multiplicity of the components of the cell rather than upon
one or two cellular components as is characteristic of prior art
methods of drug development.
[0014] It is still a further and related object of the invention to
increase the fund of knowledge relating to the interaction of a
drug candidate with multiple cellular components in order to gain
advance knowledge of the overall dynamics of the cell in the
presence of a drug candidate.
[0015] It is still a further object of the invention to provide a
method for determining in advance how a proposed drug will affect
the cell as a whole.
[0016] It is still a further object to the invention to provide
methods for simulating or analyzing the normal and disease states
of a cell and for determining how to best interact with one or more
cellular components to bring about a change in the phenotype of the
cell.
SUMMARY OF THE INVENTION
[0017] The invention is broadly in the modeling of the interactions
of the several genes, proteins and other components of a cell, the
use of mathematical techniques to represent the interrelationships
between the cell components and the manipulation of the dynamics of
the cell to determine which one or more components of a cell may be
targets for interaction with therapeutic agents.
[0018] Exemplary methods of the invention for identifying
components of a cell as putative targets for interaction with one
or more therapeutic agents, based on a cell simulation approach,
comprise the steps of:
[0019] (a) specifying a cellular biochemical network believed to be
intrinsic to a phenotype of said cell;
[0020] (b) infering new and missing links and components in the
network by using and incorporating experimental data (e.g., DNA
sequence, protein sequence, microarray, expression data, time
course expression data, protein structure, . . . etc.)
[0021] (c) simulating the network by (i) specifying its components,
and (ii) specifying interrelationships between those components and
representing the interrelationships in one or more mathematical
equations;
[0022] (d) infering new and missing links and components in the
network;
[0023] (e) constraining and or determining parameter values in the
network by (i) sampling a set of networks and parameter values,
(ii) simulating the said networks as described in (c), and (iii)
determining the network and parameter values that optimally fits a
given set or sets of experimental data (e.g., DNA sequence, protein
sequence, microarray, expression data, time course expression data,
protein structure, . . . etc.); using optimization, sensitivity
analysis, and error analysis to determine validity and robustness
of predictions
[0024] (f) solving those equations representing the network to
simulate a first state of the cell;
[0025] (g) perturbing the simulated network by deleting one or more
components thereof or changing the concentration of one or more
components thereof or modifying one or more mathematical equations
representing interrelationships between one or more of the
components;
[0026] (h) solving the equations representing the perturbed network
to simulate a second state of the cell; and
[0027] (i) comparing the first and second simulated states of the
network to identify the effect of the perturbation on the state of
the network, and thereby identifying one or more components for
interaction with one or more agents.
[0028] (j) experimentally verifying predictions from the model in
order to validate a single prediction or disceren between various
predictions or hypotheses and/or using the experimentally derived
results to iteratively refine the model.
[0029] Exemplary methods of the invention for identifying
components of a cell as putative targets for interaction with one
or more therapeutic agents based upon an analytical approach,
comprise the steps of:
[0030] (a) specifying a stable phenotype of a cell;
[0031] (b) correlating that phenotype to the state of the cell and
the role of that cellular state to its operation;
[0032] (c) specifying a cellular biochemical network believed to be
intrinsic to that phenotype;
[0033] (d) characterizing the biochemical network by
[0034] (i) identifying the components thereof, and
[0035] (ii) identifying interrelationships between the components
and representing those interrelationships in one or more
mathematical equations;
[0036] (iii) identifying new and missing links and components in
the network; and
[0037] (iv) constraining and or determining parameter values in the
network
[0038] (e) perturbing the characterized network by deleting one or
more components thereof or changing the concentration of one or
more components thereof or modifying one or more mathematical
equations representing interrelationships between one or more of
the components; and
[0039] (f) solving the equations representing the perturbed network
to determine whether the perturbation is likely to cause the
transition of the cell from one phenotype to another, and thereby
identifying one or more components for interaction with one or more
agents.
BRIEF DESCRIPTION OF THE DRAWINGS
[0040] FIG. 1 is a schematic representation of a typical gene
expression network.
[0041] FIG. 2 is a schematic representation of a typical signal
transduction/signal translation pathway.
[0042] FIG. 3 is an exemplary schematic representation, using the
Diagrammatic Cell Language, of a portion of the signal transduction
pathway and gene expression network that initiates the mammalian
cell cycle.
[0043] FIGS. 3(a) through 3(o) collectively depict the key to
reading diagrams in the format of FIG. 3.
[0044] FIG. 4 is a conventional schematic representation of the
Wnt/Beta-Catenin signaling pathway that plays a critical role in
the progression of colon cancer cells through the cell cycle.
[0045] FIG. 5 is a schematic representation of a biological network
comprising two genes.
[0046] FIG. 6 is a phase portrait for the solution of the
differential equation model for the two gene network of FIG. 5.
[0047] FIG. 7 is a graph showing the time evolution of the
differential equation model for a single copy of the gene circuit
of FIG. 5
[0048] FIG. 8a graph showing the time evolution of the stochastic
equation model for a single copy of the gene circuit of FIG. 5
[0049] FIG. 9 is a bifurcation diagram showing several states of a
cell.
[0050] FIG. 10 is a graphical representation of the Wnt
beta-catenin pathway in DCL.
[0051] FIG. 11 is a time-series profile of the concentration of
several components of a cell and represents the "normal" state of
the cell.
[0052] FIG. 12 is a time-series profile which represents the
cancerous state of the cell.
[0053] FIG. 13 is a time-series profile which shows the effect of
deleting APC from the cell.
[0054] FIG. 14 is a time-series profile which shows the effect of
deleting HDC from the cell.
[0055] FIG. 15 is a time-series profile which shows the effect of
adding Axin to the cancerous cell of FIG. 12.
[0056] FIG. 16 is a time-series profile which shows the effect of
adding HDAC to the cancerous cell of FIG. 12.
[0057] FIG. 17 is a time-series profile which shows the effect of
adding Axin and GSK3 to the cancerous cell of FIG. 12.
[0058] FIG. 18 is a time-series profile which shows the effect of
adding Axin to the cancerous cell of FIG. 12.
[0059] FIG. 19 is a time-series profile which shows the effect of
adding GSK 3 to the already perturbed cell of FIG. 18.
[0060] FIG. 20 is a time-series profile which shows the effect of
adding HDAC to the twice perturbed cell of FIG. 19.
[0061] FIG. 21 is a time-series profile which shows the effect of
reducing the concentration of Axin to zero in the normal cell of
FIG. 11.
[0062] FIG. 22 is a time-series profile which shows the effect of
reducing the concentration of GSK3 to zero in the already perturbed
cell of FIG. 21.
[0063] FIG. 23 is a time-series profile which shows the effect of
modifying the mathematical equations of the system by adding an
additional Facilitator molecule, to enhance the binding of Axin to
.beta.-catenin, to the cancerous cell depicted in FIG. 12.
[0064] FIG. 24 is a time-series profile which shows the effect of
reducing the concentration of HDAC to zero in the twice perturbed
cell of FIG. 23.
[0065] FIG. 25 is a time-series profile which shows the effect of
increasing the binding rate of Axin to b-catenin starting from the
"cancerous" cell of FIG. 12.
[0066] FIG. 26 is a time-series profile which shows the effect of
increasing the binding rate of Axin to B-Catenin slightly from the
cancerous cell of FIG. 12.
[0067] FIG. 27 is a time-series profile which shows the effect of
increasing the binding rate of B-catenin to the c-Myc TCF bound
gene from the already perturbed cell of FIG. 26.
[0068] FIG. 28 is a time-series profile which shows the effect of
increasing the binding rate of GSK3 to Axin in the twice perturbed
cell of FIG. 27.
[0069] FIG. 29 is a time-series profile which shows the effect of
setting the binding rate of Axin to GSK3 to zero in the normal cell
of FIG. 11.
[0070] FIG. 30 is a time-series profile which shows the effect of
setting the unbinding rate of B-Catenin to the C-Myc gene to zero
in the already perturbed cell of FIG. 29.
[0071] FIG. 31 is a time-series profile which shows the effect of
systematically changing parameter values to change the "cancerous"
state of FIG. 12 back to a "normal" state similar to FIG. 11.
[0072] FIG. 31(a) depicts a modular description of an exemplary
colon cancer cell simulation.
[0073] FIG. 32 depicts a modular description of an exemplary colon
cancer cell simulation.
[0074] FIGS. 32(a)-(h) depicts each module in detail so as to make
the reactions visible.
[0075] FIGS. 33(a)-(d) contain the data points and simulation for
the phosphorylated forms of AKT, MEK and ERK in the exemplary colon
cancer cell simulaiton.
[0076] FIGS. 34 and 34(a) depict the results of perturbing 50
individual targets in the exemplary colon cancer cell model.
[0077] FIG. 35 lists combinations of certain targets identified by
the exemplary colon cancer cell simulation whose absence caused
apoptosis.
[0078] FIG. 36 shows the mechanism of action of the perturbation in
the exemplary colon cancer cell simulation.
[0079] FIG. 37 depicts the simulation output of an oncogenic Ras
without autocrine signaling.
[0080] FIG. 38 depicts the simulation output of an oncogenic Ras
with autocrine signaling.
[0081] FIG. 39 depicts the simuation output of levels of Bcl2
without the G3139 antisense therapy.
[0082] FIG. 40 depicts the simulation output of inhibition of Bcl2
using the G3139 antisense therapy.
[0083] FIG. 41 depicts the simulation output of inhibiting Bec12
using G3139 antisense therapy in combination with a secondary
Chemotherapeutic agent.
[0084] FIG. 42 depicts the cleavage of PARP as a result of
inhibiting IKappab-alpha in combination with the addition of TNF at
various levels.
[0085] FIGS. 43(a)-(b) show the constructs in Diagrammatic Cell
Language.
[0086] FIGS. 44(a)-(b) compare a simple notation with the
Diagrammatic Cell Language.
[0087] FIG. 45 is a flowchart of the execution of an optimization
procedure of an exemplary software system according to the present
invention.
[0088] FIG. 46 depicts schematically one process according to the
invention for inferring a biological network.
[0089] FIG. 47 shows the network topology of the synthetic network
before and after links are removed.
[0090] FIG. 48 displays the cost to fitting the data with one link
perturbed.
[0091] FIGS. 49-50 shows the results from the exemplary network
inference methodology on a 25 node network.
DETAILED DESCRIPTION OF THE INVENTION
[0092] Definitions
[0093] The "DCL Provisional Patent" refers to that certain US
Provisional Patent Application
[0094] "Cancer" and "Disease" have their usual meanings and may be
used interchangably.
[0095] "Equation" refers to a general formula of any type or
description and also includes computer code and computer readable
and/or executable insturctions;
[0096] "Formulae" and "Equations" have their normal meanings and
are used interchangably and without limitation.
[0097] "Phenotype" of a cell means the detectable traits of a cell,
i.e. its physical and chemical characteristics, as influenced by
its environment;
[0098] "State of a cell" means, in the aggregate, the components of
the cell, the concentrations thereof and their interactions and
interrelationships;
[0099] "Cellular biochemical network" means a subset of the
components of the cell and their known or posited interactions and
interrelationships;
[0100] "receptor" a site on a cell (often on a membrane) that can
combine with a specific type of molecule to alter the cell's
function
[0101] "EGF" refers to Epidermal Growth Factor
[0102] "EGFR" refers to Epidermal Growth Factor Receptor
[0103] "Erk" refers to a kinase in the Ras Map Kinase cascade
[0104] "Functional output" refers to an output of a simulation
which is a function of time, such as, e.g., a time series for a
given biochemical;
[0105] "Intrinsic to said phenotype" means causing or contributing
to the phenotype;
[0106] "Mek" refers to a well known kinase in the Ras Map Kinase
cascade
[0107] "NGF" refers to Nerve Growth Factor
[0108] "NGFR" refers to Nerve Growth Factor Receptor
[0109] "Parameters" refer to any biochemical network component
(such as e.g., chemicals, protiens, genes, rate constants, initial
concentrations, etc.) that can vary that can change the final
output of a biochemical network;
[0110] "Putative target for interaction" means, broadly, any
cellular component whose existence or concentration is determined,
by practice of the methods of the invention, to have a significant
effect on the phenotype of the cell such that when removed from the
cell or reduced or increased in concentration, the phenotype may be
altered. More specifically, a "putative target for interaction"
means a cellular component which appears to be an actual physical
or chemical target for a binding agent or reactant which will have
the effect of removing the target or changing its
concentration;
[0111] "Raf" is a kinase in the Ras Map kinase pathway
[0112] "Ras" is a small G-protein implicated in over 40% of all
cancers;
[0113] "Attractors of a cell" are asymptotical dynamic states of a
system. These are the fixed points, limit cycles, and other stable
states that the cell tends to as a result of its normal behavior,
an experimental perturbation, or the onset of a disease.
[0114] "degradation" is the destruction of a molecule into its
components. The breaking down of large molecules into smaller
ones.
[0115] "Ubiquitination" refers to the process involving A 76-amino
acid polypeptide that latches onto a cellular protein right before
that protein is broken down
[0116] "endocytosis" is the process by which extracellular
materials are taken up by a cell
[0117] "signal transduction" refers to the biochemical events that
conduct the signal of a hormone or growth factor from the cell
exterior, through the cell membrane, and into the cytoplasm. This
involves a number of molecules, including receptors, proteins, and
messengers
[0118] "transcription" the synthesis of an RNA copy from a sequence
of DNA (a gene); the first step in gene expression
[0119] "translation" The process in which the genetic code carried
by messenger RNA directs the synthesis of proteins from amino
acids
[0120] "G1" refers to the period during interphase in the cell
cycle between mitosis and the S phase (when DNA is replicated).
Also known as the "decision" period of the cell, because the cell
"decides" to divide when it enters the S phase. The "G" stands for
gap.
[0121] "S phase" refers to the period during interphase in the cell
cycle when DNA is replicated in the cell nucleus. The "S" stands
for synthesis.
[0122] "G2" refers to the period during interphase in the cell
cycle between the S phase (when DNA is replicated) and mitosis
(when the nucleus, then cell, divides). At this time, the cell
checks the accuracy of DNA replication and prepares for mitosis.
The "G" stands for gap.
[0123] "mitosis or M phase" refers to the process of nuclear
division in eukaryotic cells that produces two daughter cells from
one mother cell, all of which are genetically identical to each
other.
[0124] "apoptosis" refers to programed cell death as signalled by
the nuclei in normally functioning human and animal cells when age
or state of cell health and condition dictates. Cancerous cells,
however, are unable to experience the normal cell transduction or
apoptosis-driven natural cell death process
[0125] "cleavage" refers to the breaking of bonds between the
component units of a macromolecule, for example amino acids in a
polypeptide or nucleotide bases in a strand of DNA or RNA, usually
by the action of enzymes
[0126] oligomerization refers to the chemical process of creating
oligomers from larger or smaller molecules.
[0127] "mitochondrion" is an organelle found in eukaryotes
responsible for the oxidation of energy-rich substances. They are
oval and have a diameter of approximately 1.5 micrometers and width
of 2 to 8 micrometers. Mitochondria have their own DNA and are
thought to have evolved when an early eukaryote engulfed some
primitive bacteria, but instead of digesting them, harnessed them
to produce energy.
[0128] cytochrome c is a type of cytochrome, a protein which
carries electrons, that is central to the process of respiration in
mitochondria (an organelle found in eukaryotes which produces
energy).
[0129] Analysis of Genomic and Cellular Information
[0130] In recent years, efforts have been made to harness the
information becoming available from the Human Genome Project and
other information relating to cellular dynamics. Information at the
genetic level determines the form and function of a cell or
organism. The information contained in the DNA sequences of the
genes of an organism, the genotype, is expressed to determine the
phenotype, the state of the cell or organism. With the completion
of the Human Genome Project, the goal of predicting how the
phenotype of a biological system arises from the information
encoded in the genotype has been made more achievable. The
challenge for molecular medicine is to understand how particular
changes or mutations in the genes lead to the onset of disease and
to determine the best strategy for reversing the disease
phenotype.
[0131] To predict and understand mechanistically how the phenotypes
of a cell arise from the gene sequences requires an understanding
of biochemical networks. These complex networks consist of genes
and proteins that control how specific genes are expressed in
response to a cell's current state and its environment.
[0132] Gene Expression Networks
[0133] FIG. 1 is a schematic representation of a gene expression
network. It is well understood that when particular regulatory
proteins and transcription factors are bound to the promoter
sequence on a gene, the expression of mRNA molecules of the gene is
turned `on` or `off`. During gene expression, RNA polymerase
"reads" the DNA sequence of the gene to transcribe it, i.e., to
produce a specific mRNA molecule. This specific mRNA molecule is in
turn decoded by a ribosome that translates the mRNA, i.e., creates
a specific protein.
[0134] Proteins control metabolism, response to environmental cues
and regulation of other genes. Regulatory proteins bind to the
promoter sequences, which act as a switch to regulate the
expression of a nearby gene. FIG. 1 shows gene 1 producing protein
1, which binds to the promoter region of gene 2, activating the
expression of gene 2. Gene 2 then produces protein 2, which then
binds to the promoter region of gene 3, turning off the expression
of gene 3. If gene 3 is active, it will produce a protein that
binds to the promoter region of gene 1, which then activates the
expression of gene 1. This series of gene-to-protein and
protein-to-gene interactions represents a gene expression network.
Such networks ultimately control the overall levels of gene
expression for the entire genome and consequently determine the
phenotypes of the cell. Signal Transduction Pathways Signal
transduction pathways are another important class of biochemical
networks. These pathways communicate information about the
environment outside of a cell to the genes inside the nucleus of
the cell. FIG. 2 is a schematic representation of a typical signal
transduction/translation pathway. Genes 4, 5, 6 and 7 produce
proteins that reside in the cytoplasm of the cell. Gene 4 codes for
a receptor, a signal transduction protein that lies embedded in the
membrane of the cell, with one part on the outside facing the
environment and the other part on the inside facing the cytoplasm.
Growth factors, hormones, and other extracellular signals bind to
receptors and activate a cascade of biochemical reactions. The
proteins involved in signal transduction pathways, including
receptors, are allosteric; i.e., they exist in an inactive and an
active state. Biochemical reactions such as phosphorylation of a
particular part of a protein or the exchange of a bound GDP
molecule for a GTP molecule can change the state of an allosteric
protein from inactive to active. Once activated, these proteins
bind to or react with other proteins to activate them. Signal
transduction pathways are thus activated in a domino-like fashion.
A signal at the cell surface from a receptor binding event starts a
cascade of biochemical reactions and information flow which leads
to the transport of a particular protein into the nucleus where it
then activates transcription factors that in turn activate the
expression of one or more genes. Signal transduction pathways,
which transmit information from outside the cell to the genes, are
thereby coupled to the gene expression networks that control the
expression patterns of the genes and the state of a cell, i.e. its
phenotype.
[0135] Disease State Prediction by Modeling Biochemical
Networks
[0136] Biochemical networks and their interactions with the
environment ultimately determine the state of a cell and the
development of a disease state. Deciphering the complex intertwined
gene expression and signal transduction pathways involving hundreds
or thousands of molecules has proven difficult. To predict and
understand how mutations, particularly in genes, lead to a disease
state, e.g. the development of cancer, requires predicting the
behavior of these networks. This necessitates the formulation of
mathematical equations that quantitatively describe how the
concentrations of gene products, mRNA, inactive and active
proteins, change in time in response to extracellular signals such
as growth factors and hormones and to the concentrations of such
products of other genes.
[0137] The equation for the time rate of change of a particular
protein or mRNA is comprised of terms derived from enzyme kinetics.
These terms describe reactions that create, destroy, or modify the
protein, i.e. phosphorylation, dephosphorylation, translation and
degradation reactions. The differential equations are nonlinear,
but they can be solved analytically or by computer simulation to
produce a plot of the concentrations of mRNA and protein as a
function of time. The time series of concentrations for any
particular mRNA or protein can be high, intermediate, or low or can
oscillate in time or even change chaotically in time, Hill et al,
Proc. of Statistical Mechanics of Biocomplexity, Springer-Verlag
(1999). A particular time series profile corresponds to a
particular state of gene expression and thus to a particular
biological state. The difference between a normal and a cancerous
state manifests itself at this level of description. A particular
time series of concentrations corresponds to a healthy cell whereas
another time series of concentrations corresponds to a disease
state, e.g. cancer.
[0138] Previous mathematical modeling has only been applied to
relatively simple networks, such as lysis-lysogeny in E. coli,
McAdams and Shapiro, Science, 650, 1995, the lac operon in E. coli,
Wong et al, Biotechnol. Prog., 132, (1997), and circadian rhythms
in Drosophila, Tyson et al., Biophysical Journal, 77:2411, 1999.
Some of these models are based on chemical rate equations. A few
emphasize the key role played by stochastic fluctuations due to the
small numbers of molecules in a given cell, McAdams and Arkin,
Proc. Natl. Acad. Sci. USA, 814, 1998. Boolean switching networks,
nonlinear and piecewise linear differential equations, stochastic
differential equations and stochastic Markov jump processes all
provide mathematical frameworks that can represent the time
evolution of mRNA and protein concentrations. The kinetic equations
for the mRNA and proteins involved in biochemical networks are
solved in simple cases (fewer than three genes) with analytical
methods from nonlinear dynamics (bifurcation analysis, linear
stability analysis, etc.) and statistical physics and probability
theory (master equation, stochastic calculus, methods of stochastic
averaging). In the more general, high dimensional cases, it is
sometimes possible to extend these analytical methods and to use
object-oriented computer simulations for deterministic dynamics
(Runge-Kutta integration) and stochastic dynamics (Monte Carlo
simulation), Gillespie, J. of Comp. Phys. These techniques can make
the problem of understanding complex gene expression networks and
signal transduction pathways tractable.
[0139] A number of difficulties confront researchers who propose to
use mathematical and computational frameworks to predict disease
states. Among these are the ill-defined nature of networks of
interaction, incomplete forms of kinetic equations, incomplete
mathematical frameworks, the absence of quantitative measurements
of gene product concentrations and the absence of quantitative
measurements of reaction rate constants and other kinetic
parameters.
[0140] Descriptions in the literature of biochemical networks which
determine cell phenotypes is incomplete and developing slowly.
Protein-protein and protein-gene interactions are detected mainly
through immunoprecipitation techniques and footprinting methods.
Since only a relatively small number of biological phenomena have
well-defined biochemical and genetic circuits, detailed pictures of
the molecular interactions in systems such as the mammalian cell
cycle, have only begun to emerge over the last decades.
[0141] Yeast two-hybrid experiments for "fishing out" the binding
partner of a protein have been employed in parallel on the genome
scale to find such protein-protein interactions, G. Church,
Harvard; Curagen. Methods of computationally mining the genome have
also been employed to predict unknown protein-protein interactions,
Eisenberg, UCLA; Protein Pathways, Inc. Methods have also been
developed to find transcription factors (regulatory proteins) and
their corresponding cis-acting binding sites, G. Church, Harvard.
In time, these academic and industrial genome-wide efforts will
uncover many new biochemical networks and fill in the missing links
in more well-defined systems.
[0142] Many equations describing the rates of enzyme-catalyzed
reactions have been derived empirically, often without rigorous
theoretical justification. Despite an increase in the understanding
of the mechanisms of many cellular processes (e.g., the discovery
of scaffolds, reactions on membranes and active transport), a
reformulation and extension of the fundamental kinetic forms used
to describe the chemical reactions controlling cellular behavior is
not yet available. Some information may become available in the
future as biophysicists and biochemists elucidate the molecular
mechanisms for many processes, e.g. molecular motors, chromatin
structure dynamics, vesicle transport and organelle formation.
[0143] Different kinetic forms of biochemical equations can be
expressed within several mathematical frameworks that have been
developed to model gene networks. All such mathematical frameworks
are approximations to reality. Modeling efforts began with
nonlinear differential equations, then Boolean approximations, and,
more recently, with stochastic formulations. Most such efforts
assume that the spatial extent of the cell is not important, thus
ignoring diffusion processes. There is still no consensus on what
frameworks should be used, and few rigorous results have been
reported. More recently, researchers have begun studying stochastic
formulations of gene expression. Theoretical progress has been made
in understanding the effects that `noise` can have on biological
systems and in determining which mathematical frameworks should be
used in which contexts. The recent creation of artificial gene
networks also provides a testing ground for some of these
theoretical results, Gardener et al, Nature (2000); Elowitz and
Leibler, Nature (2000). New techniques such as fluorescence and
imaging for dynamically monitoring the expression and activity of
biomolecules provide accurate data with which to compare
mathematical predictions.
[0144] Measurements of the concentrations of proteins and mRNA have
rarely been reported in the literature, as biologists have
primarily focused their efforts on qualitative observations rather
than on quantitative measurements. However, mathematical models of
gene networks which can predict a time series of proteins and mRNA
concentrations require accurate quantitative concentration
measurements of protein and mRNA over time. Without such
information it is difficult to validate any predictions from the
model. With the recent development of DNA microarrays, it is now
possible to monitor the genome-wide concentrations of all mRNA
species in a single experiment and to view the state of the cell as
determined by mRNA concentrations. Imaging techniques and
proteomics also provide in vivo snapshots of protein concentrations
and localization.
[0145] As well, quantitative measurements of rate constants are
also rarely reported. As a result, the measurement of rate
constants and other kinetic parameters lags behind the knowledge of
the organization of many protein and gene networks. These rate
parameters in a mathematical model ultimately determine the model's
predictions and are thus of paramount in importance. Changes in the
rate parameters define the onset of disease.
[0146] The Biochemical Foundations of Cancer
[0147] Cells become cancerous when several genes are mutated and
their protein products cannot function normally. This dysfunction
causes the highly regulated cell cycle machinery to break down,
leading to uncontrolled cell growth. The transformation of a normal
cell to a cancerouscell is a multi-step process involving a complex
biochemical network or networks involving hundreds of genes and
proteins. In such transformations several genes are mutated, one
after another, often in a particular order. Each of these mutations
causes morphological and physiological changes (see Vogelstein
1995). It has also become increasingly apparent that cancer is
often caused by a combination of cooperating oncogenes, none of
which is dominant. This complex combinatorial genetic origin makes
genotype-to-phenotype mapping a difficult problem, and one that
must be understood better before rational approaches to cancer
chemotherapy can be achieved. Because these genes interact within
the gene network to collectively cause transformation to a
cancerous state, a mathematical description is required to identify
the combination of interacting genes that can reverse or stop
uncontrolled cell growth.
[0148] While many of the mutated genes that lead to cancer have
been identified, because of the complexities in understanding how
and in what combinations these genes actually precipitate a cancer,
a systematic and quantitative description of the networks involved
is required to better understand the cellular dynamics of the
disease.
[0149] The Mammalian Cell Cycle
[0150] The methods of the invention can be broadly applied to find
targets for therapeutic agents among the many components of the
mammalian cell cycle. The invention is exemplarly described below
with respect to one portion of that cycle.
[0151] FIG. 3 depicts a portion of the signal transduction pathway
and gene expression network that initiates the mammalian cell
cycle. The several gene and protein components of the network are
identified and described below.
[0152] FIG. 3 was created using the using Diagrammatic Cell
Language ("DCL"), a computer based graphic language which has been
devised to describe all of the interactions in a cell, or within a
particular biochemical network, in a single diagram, with only a
few representations of each molecule. The notation is explained in
detail in the article entitled "Dramatic Notation and Computational
Structure of Gene Networks" by Ron Maimon and Sam Browning, which
can be found at www.gnsbiotech.com (the website maintained by the
assignee hereof). DCL is a novel means for facilitating interaction
between biology and quantitative methods or applied mathematics to
biology/biochemistry. The objects available in DCL for modeling
chemicals, protiens, genes, and other components of cells,
subcellular biochemical netoworks, biochemical pathways, or even
virtual biochemical networks involving interactions between
numerous cells in variant cell populations, each have built in
associated quantitative mathematical expressions. Thus, when a
biologist or other biochemical network modeler constructs a model
of some cellular or subcellular network in the DCL environment
using the objects available in DCL, she need not know the precise
mathematical expression of these objects. Nonetheless, the DCL
parser can take the constructed model and generate a precise
mathematical description of the modeled biochemical network, such
that it can be solved, optimized and perturbed according to the
methods of the present invention.
[0153] By analogy to microelectronics, DCL brings to the biological
sciences the equivalent functionalities of SPICE, the well known
microelectronic circuit modeling tool. A key to interpreting the
DCL language or notation symbology is summarized in FIGS. 3(a)
through 3(n).
[0154] The box shown in FIG. 3(a) is used to represent a chemical
which is a single indivisible chemical unit. Examples are seen at
numerous locations in FIG. 3.
[0155] FIG. 3(b) shows reversible binding between chemicals A and
B. C is a component which is stimulating the unbinding of A and
B.
[0156] The symbol in FIG. 3(c) indicates irreversible binding. FIG.
3(d) shows a link box which is used to indicate components in the
cell with complex structures, such as a protein or DNA. A link box
can contain other objects, such as binding nodes (the solid black
circles in FIG. 3(d)) which represent functional binding sites on a
protein. The link box shown in FIG. 3(d) can, for example, bind the
two chemical substances A and B. The unbinding symbol is shown in
FIG. (1).
[0157] The numbers in parenthesis (RES) in the line leading from
the Link Box indicate the resolution of the states of the
substances in the link box. For example, the numbers shown (0,1)
indicate that component A is not bound, and component B is
bound.
[0158] FIG. 3(e) shows an internal link box L. This is used to
identify a particular state similar to the resolution shown in FIG.
3(d). Here, it is shown that the state of box A and B bound (i.e.,
the dimer comprised of A and B) is chosen to interact with the
other entities in the cell.
[0159] The combination of boxes in FIG. 3(f) is called a Like Box.
This is used to depict which groups of objects are alike in
functionality. Various components within the box also can be
resolved to choose a particular state. The resolution indication
RES on the line emerging from the large box indicates a state of 1,
or component B regulating other cellular or network entities.
[0160] In FIG. 3(g) the wavy line R indicates a reversible
reaction. In the example shown, A and B are involved in a
reversible reaction R to produce C. E is an enzyme driving the
reaction towards the product D.
[0161] FIG. 3(h) indicates an irreversible reaction, with a
characteristic one-way arrow.
[0162] FIGS. 3(i) through 3(n) are self-explanatory, illustrating,
respectively, an unbinding stimulation; a binding stimulation; an
enablement; no reaction, an enhancement, and a directional
stimulation.
[0163] With refernce to FIG. 3, the components of the network
interact as follows. The binding of epidermal growth factor (EGF)
301 or nerve growth factor (NGF) 302 to its respective receptor
(EGFR 303, NGFR 304) results in the activation of SOS 305, 305A, a
guanine nucleotide exchange factor. SOS then dislodges the GDP
molecule from Ras 306, 306A a small G-protein implicated in over
40% of all cancers. When the GDP molecule is dislodged from Ras,
Ras then binds a GTP molecule, which shifts Ras into its active
form (Ras+ 306A). Ras+ 306A then activates the Map kinase cascade
consisting of Raf 307, Mek 308, and Erk 309. When Erk 309 is
activated, it is transported to the nucleus where it activates the
transcription factor Sp1 310. Sp1 310 then activates the
transcription of some of the important cell cycle genes such as p21
320 and cyclin D 321 that drive the cell to replicate its DNA and
to ultimately divide into two daughter cells.
[0164] While an overview of cell cycle control and cellular
response to environmental cues is available, Kohn, Molec. Biol.
Cell. 10: 2703 1999., available on the World Wide Web at URL
http://discover.nci.nih.gov/kohnk/- interaction_maps.html, there
are several examples where networks of genes and gene products
respond to cues in ways that are not obvious based on a knowledge
of the network constituents. Signal transduction pathways once
thought to act independently in determining cell fate
(proliferation, apoptosis and differentiation) have been found to
interact at a number of molecular "nodes", See Bhalla and Tyengar
1999. This gives rise to "crosstalk" and feedback loops between
signal transduction pathways and the genes and gene products that
they control (e.g., Ras, p21, etc.), as described by Kohn, Molec.
Biol. Cell. This crosstalk leads to unexpected biological outcomes
and requires a more sophisticated and systematic description, such
as is provided by the present invention. Similarly, differences in
both the levels and durations of activation of certain gene
products (e.g. Ras or Erk), see Joneson and Bar-Sagi, Traverse et
al. 1992, have been shown to result in very different cellular
responses. Thus a quantitative predictive model for the relevant
interactions is required to make any meaningful predictions.
Moreover, due to the complexity of nearly all biochemical networks,
to be useful, a quantitative method needs to be capable of managing
large scale, multinodal, interconnected systems. Such a
quantitative model is provided by the present invention.
[0165] The methods and implementations of the present invention are
used to discover targets for therapeutic agents by predictions from
simulation studies and analytical studies. The methods are
supported by mathematical techniques which infer relationships
among the various components of a biochemical network.
[0166] With complete and accurate information about the biochemical
networks being studied, the simulation and analytical studies
provide accurate predictions about the behavior of the system and
the identity of the targets. Optimization techniques can be used to
constrain uncertainty inherent in the use of large genomics data
sets. While the simulation and analytical studies are powerful
given `perfect data` as inputs to the models, `perfect data` does
not exist. Therefore, data mining techniques and bioinformatics
from the analysis of large data sets of DNA sequence and expression
profiles are used to provide meaningful correlations and patterns
among the network components. The underlying structures that data
mining attempts to locate, such as markers and partitions between
normal and cancerous cells, are ultimately a manifestation of the
underlying dynamics of the biochemical network. Recent studies on
differential gene expression reveal genes that are misregulated in
disease states. These genes are potential targets and are used in
computer models according to the present invention. As well,
pattern recognition algorithms and artificial intelligence methods
are used to elucidate subtle relationships among genes and proteins
as well as to uncover the underlying data structure, e.g.
partitions between cell types, cancer types and stages of
malignancy. The combination of the predictive and the inferential
approaches leads to the discovery of multiple targets.
[0167] Simulation Embodiments of the Invention
[0168] For ease of illustration purposes, the present invention
will often be described herein in terms of a cell or a biochemical
network within a cell. This is for exemplary purposes only, and is
not intended to limit the application soft represent invention. The
term cell is thus intended to include biochemical networks of any
type, now known or which are as yet unkonwn, some cellular, some
subcellular, and some supercellular.
[0169] One or more components of a cell or other biochemical
network may be identified as putative targets for interaction with
one or more therapeutic agents by performing a method comprising
the steps of:
[0170] (a) specifying a cellular biochemical network believed to be
intrinsic to a phenotype of said cell;
[0171] (b)simulating the network by
[0172] (i) specifying its components, and
[0173] (ii) specifying interrelationships between those components
and representing the interrelationships in one or more mathematical
equations;
[0174] (c) solving those equations to simulate a first state of the
cell;
[0175] (d) perturbing the simulated network by deleting one or more
components thereof or changing the concentration of one-or more
components thereof or modifying one or more mathematical equations
representing interrelationships between one or more of the
components;
[0176] (e) solving the equations representing the perturbed network
to simulate a second state of the cell; and
[0177] (f) comparing the first and second simulated states of the
network to identify the effect of the perturbation on the state of
the network, and thereby identifying one or more components for
interaction with one or more agents.
[0178] As well, after each of steps (c) and (e) above, an
additional optimization step could be performed, where the solution
of the mathematical equations simulating a state of the cell is
optimized to have minimum error vis-a-vis the prediction of certain
available experimental data. Such optimization would also comprise
error analysis and extrapolation emthods. Specific methods of
optimization and handling of error are described more fully
below.
[0179] The mathematical equations representing the
interrelationships between the components of the cellular
biochemical network are solved using a variety of methods,
including stochastic or differential equations, and/or a hybrid
solution using both stochastic methods and differential equations.
In carrying out the methods of the invention, the concentrations of
one or more of the several proteins or genes (generally
"components") in the biochemical network are selectively perturbed
to identify which ones of those proteins, genes or other components
cause a change in the time course of the concentration of a protein
or gene implicated in a disease state of the cell.
[0180] Thus, a series of perturbations are made, each of which
changes the concentration of a protein, gene or other component in
the network to a perturbed value. The mathematical equations are
then solved with stochastic or differential equations (or some
hybrid thereof) to determine whether that protein or gene is
implicated in causing a change in the time course and/or spatial
localization of the concentration of a protein or gene implicated
in a disease state of the cell. In particular embodiments of the
invention, the concentration of each of the proteins and genes in
the network is reduced to zero in each respective perturbation.
[0181] Again referring to FIG. 3, it is known, for example, that
the presence of NGF 302 in the network causes the cell to
differentiate whereas the presence of EGF 301 in the network causes
the cell to proliferate, a condition which may lead to the
development of cancer. It is known that the concentrations of NGF
302 and EGF 301 affect the time course of the concentration of Erk
309, NGF 302 causing Erk 309 to increase with time and ultimately
reach a steady-state value and EGF 301 causing the concentration of
Erk 309 to initially increase and then decrease to a lower level
than is present with NGF 302 in the network.
[0182] In the example network depicted in FIG. 3, to which an
exemplary embodiment of the invention is applied, it is desirable
to convert EGF 301, which causes the cell to proliferate, into NGF
302, which causes the cell to differentiate. The system is
manipulated by perturbing it to block out, inhibit, activate or
overactivate one or more of the components of the network to see if
that component is implicated in causing the time course of NGF 302
to increase and the time course of EGF 301 to decrease. The time
courses of these components are believed to be a surrogate for
predicting whether the cell will proliferate or differentiate and
consequently whether the cell will become cancerous or not.
[0183] In this particular example, using the methods and
implementation of the invention, the concentration of the several
components of the network set forth in FIG. 3 were each
respectively reduced to zero in a separate perturbation and the
time course of the concentrations of NGF and EGF were determined
from calculations of the concentrations of cellular components
according to the simulation. It was found that when component P13K
was "knocked out", i.e., when its concentration was reduced to
zero, this caused the time course of the concentration of NGF to
increase and the time course of the concentration of EGF to
decrease, indicating a beneficial result in that the cell was,
according to this surrogate analysis, caused to differentiate
rather than to proliferate. Thus, P13K was identified as a target
which when removed from the cellular network will cause the
beneficial result described above.
[0184] Analytical Embodiments of the Invention
[0185] In another embodiments of the invention, one or more
components of a cell (in the general sense, as described above) may
be identified as putative targets for interaction with one or more
therapeutic agents by performing a method comprising the steps
of:
[0186] (a) specifying a stable phenotype of a cell;
[0187] (b) correlating that phenotype to the state of the cell and
the role of that cellular state to its operation;
[0188] (c) specifying a cellular biochemical network believed to be
intrinsic to that phenotype;
[0189] (d) characterizing the biochemical network by
[0190] (i) specifying the components thereof, and
[0191] (ii) specifying interrelationships between the components
and representing those interrelationships in one or more
mathematical equations;
[0192] (e) perturbing the characterized network by deleting one or
more components thereof or changing the concentration of one or
more components thereof or modifying one or more mathematical
equations representing interrelationships between one or more of
the components; and
[0193] (f) solving the equations representing the perturbed network
to determine whether the perturbation is likely to cause the
transition of the cell from one phenotype to another, and thereby
identifying one or more components for interaction with one or more
agents.
[0194] In analytical method embodiments of the invention, the
object is not to find numerical values of the several components of
the biochemical network, as described above in performing the
methods of the invention in a simulated network. In the analytical
embodiments, instabilities and transitions of a cell state are
identified and a bifurcation analysis is conducted in order to
analyze the stability of the cell and determine the probability of
its transformation into a different state, e.g. a disease state.
Such an exemplary bifurcation analysis is depicted in FIG. 9, which
is a bifurcation diagram plotting various cell phenotypes as a
function of two rate constants (Rate Constant 1 ("RC1") and Rate
Constant 2 ("RC2") in FIG. 9). In FIG. 9, a phenotype of normal
cell growth is seen when RC1 is low and RC2 is high. Cancer is seen
when RC1 and RC2 are both high, apoptosis when RC1 is high and RC2
is low, and differentiation when both RC1 and RC2 are low.
[0195] In an analytical method embodiment of the invention,
attractors of the system are identified. These attractors are the
equilibrium states of the cell. Attractors may be steady state
equilibria, periodically changing equilibria or chaotic equilibria
with certain peculiar signatures, of the network. The equilibria
may represent normal, disease, growth, apoptosis or other states of
the cell, as depicted n the example of FIG. 9.
[0196] Once these attractors have been identified, it is possible
to perturb the network to determine conditions under which the cell
may be transformed from one state to another. Employing stochastic
calculus, one may calculate the size of fluctuations around these
fixed-point attractors. This permits a determination of the
probability of a transformation from one state to another and the
length of time of such a transformation and identifies which
proteins and/or genes contribute to the stochastic fluctuations. It
is also possible to study the degree of stimulation and the
duration of stimulation required to move a cell from one biological
state to another. It is known that strong stimulation for an
extended period of time, e.g. from a growth factor, can "push" a
cell from a normal state to a cancerous state. Prolonged stimulus
can also deactivate the cell into a stable state. All of these
dynamics lead to the identification of drug targets as well as
information of value with respect to the duration of a drug
treatment that may be needed.
[0197] The analytical embodiments of the invention use root
findings and continuation algorithms to find bifurcations rather
than conducting repeated simulations as in the simulation
embodiments. Understanding which parameters and which proteins and
genes are important in causing or reversing a cancerous state may
lead to the identification of multiple-site drug targets.
[0198] It is also possible using the methods of the invention to
evolve the biochemical network. By elucidating the connections
between the components of the network and their functional
dynamics, the methods of the invention lead to a prediction of the
changes which may give rise to disease or which may cause a disease
state to transition into a normal state. It is also possible to
find and "evolve" states that are more stable than the starting
state of the network. These evolved states can be experimentally
checked to see if the biology is accurately described by the
predicted state of the network. Considering evolved networks also
helps to rule out poor drug targets. Cancer cells are frequently
subject to high mutation rates and a treatment that is predicted by
the method of the invention and that is robust in the face of
evolved changes in the network will be more desirable than
treatments leading to a changed state that is easily sidestepped by
minor evolutionary changes in the cell.
[0199] In performing the methods of analytical embodiments of the
invention, the concentrations of one or more of the several
proteins and genes in the biochemical network are selectively
perturbed to identify which ones of those proteins or genes are
implicated in causing an attractor of the biochemical network to
become unstable. Thus a series of perturbations are made which
change the concentration of a protein or gene in the network to a
perturbed value. Solving the mathematical equations representing
the interrelationships between the components of the network then
leads to a determination of whether the perturbed protein or gene
is implicated in causing a change in the time course of the
concentration of a protein or gene implicated in a disease state of
the cell. In preferred analytical embodiments of the invention, the
concentrations of each of the proteins and genes is reduced to zero
in each respective perturbation and the mathematical equations are
then solved.
[0200] In particular preferred analytical embodiments of the
invention, a bifurcation analysis is performed using eigenvalues of
a Jacobian matrix based upon the equations describing the
interrelationship of network components to characterize the
stability of one or more attractors.
[0201] Identification and Selection of Biochemical Networks of
Disease
[0202] The literature provides sources for identification of
biochemical networks intrinsic to disease studies. These networks
include signal transduction pathways governing the cell cycle;
transcription, translation, and transport processes governing the
cell cycle; protein-gene and protein-protein interactions, such as,
e.g., Kohn maps; protein-protein interactions found through genome
data mining techniques; protein-protein interactions from
genome-wide yeast-two hybrid methods; trans-acting regulatory
proteins or transcription factors and cis-acting binding motifs
found through experimental and computational genome-wide search
methods; protein-gene and protein-protein interactions inferred
through the use of microarray and proteomics data; protein-protein
interactions and protein function found from 3-dimensional protein
structure information on a genome-wide scale; binding partners and
functions for novel uncharacterized human genes found through
sequence homology search methods; and protein-protein interactions
found from binding motifs in the gene sequence.
[0203] The Biochemical Network Involved in Colon Cancer
[0204] FIG. 4 describes the Wnt/Beta-Catenin signaling pathway that
plays a critical role in the progression of colon cancer cells
through the cell cycle. FIG. 4 uses a conventional depiction, and
comes from Science Magazine, Signal Transduction Knowledge
Environment web page at
http://stke.sciencemag.org/cgi/cm/CMP.sub.--5533. The various
letters labelling the genes and proteins in FIG. 4 indicate the
location of the molecules specified in the legend with reference to
the cell, where "E" means extracellular, "P" means Plasma membrane,
"C" means Cytosol, "N" means Nucleus, and "O" means Other
Organelle. The symbols +, -, and 0 indicate the type of interaction
between the molecules. A detailed explanation of this pathway is
found at the website listed above.
[0205] Other subnetworks which can be examined using the methods of
the invention, include, for example, (1) Ras-Map Kinase pathway,
(2) Wnt/f-Catenin, (3) G1-S transition, (4) Rho-family G proteins
(cdc42, etc.), (5) JNK pathway, (6) Apoptosis (Caspases, p53), (7)
G2-M transition, (8) Integrin pathway, (9) P13 Kinase pathway, (10)
c-Myc pathway, (11) Telomeres, (12) Nuclear Receptors, and (13)
Calcium Oscillations.
[0206] Kinetic and Expression Data
[0207] The creation of quantitative and accurate mathematical
models of biochemical networks requires knowledge of all kinetic
parameters involved in the network. Concentrations of mRNA and
protein as a function of time provide necessary data for
optimization routines to locate meaningful values of kinetic
parameters. Kinetic parameters such as reaction rate constants,
equilibrium constants and expression data and mRNA and protein
concentrations for the gene products involved, are found in the
literature. Kinetic and expression data are available from
literature searches, public databases such as the National Cancer
Institute Cancer Anatomy Project, private sources and experimental
data. In addition to kinetic parameters and expression data, data
on the localization of mRNA and proteins, and the structure and
function of molecules may be used.
[0208] Mathematical Models of Biochemical Networks
[0209] The following equations represent the interrelationships
among the components of the biochemical network of FIG. 3. The
equations are in terms of more general descriptors of the
components of FIG. 3, and thus the general term "FreeReceptor" in
the equations relates to EGFR, and the term "Protein" relates to
EGFLigand. As well RasGTP appears as Ras+in the figure, and an
asterisk in the equations denotes a "+" in FIG. 3 (refrring to the
phosphorylated, or activated form of the component). 1 [
FreeReceptor ] t = - k b [ FreeReceptor ] [ Protein ] + k u [
FreeReceptor : Protein ] [ RasGTP ] t = k phos [ SOS * ] [ RasGDP ]
K m + [ RasGDP ] - k dephos [ RasGAP ] [ RasGTP ] K md + [ RasGTP ]
[ Mek * ] t = k phos [ Raf1 * ] Mek K m + Mek - k dephos [ Mek * ]
K md + [ Mek * ] [ FreePromoter ] t = - k b [ FreePromoter ] [
Protein ] + k u [ FreePromoter : Protein ]
[0210] The equations quantitatively describe the time rate of
change of gene products (mRNA, inactive protein, and active
protein) that comprise the biochemical network. Each term in such
differential equations represents a particular reaction in the
biochemical network. A particular reaction in the biochemical
network is represented in FIG. 3 by an arrow connecting two or more
biomolecules.
[0211] The form of each of these terms is derived through the
fundamental relations of enzyme kinetics. For enzyme-catalyzed
reactions that satisfy certain criteria, these terms are nonlinear
functions of the concentration of inactive substrate: the
Michaelis-Menten forms. The differential equation for RasGTP
contains a term describing the SOS catalyzed conversion of RasGDP
to RasGTP. This first term indicates that the rate of creation of
RasGTP is equal to the product of a rate constant, Kphos, the
concentration of active SOS and the nonlinear function of the
inactive substrate concentration RasGDP. The nonlinear activation
of RasGTP causes the rate of activation to saturate at high levels
of inactive substrate. This term indicates that RasGTP is rapidly
produced when there is a high concentration of active SOS, is
slowly produced when there is little active SOS, and is not
produced at all when either active SOS or RasGDP concentrations are
zero. A similar term describes the enzyme-catalyzed deactivation of
RasGTP by RasGTP.
[0212] The kinetic form for each reaction often varies with
reaction type. For example, in the differential equation describing
the rate of change of free promoter concentration, the term
describing the binding of free promoter to protein is the product
of the binding rate, the free promoter concentration, and the
protein concentration.
[0213] Mathematical frameworks other than nonlinear differential
equations can be used to describe the dynamics of biochemical
networks. When certain assumptions are made about the enzyme
catalyzed Michaelis-Menten reaction form, the nonlinear term
becomes piece-wise linear or `switch-like` and is more amenable to
mathematical analysis. Kauffman 1969, Kauffman and Glass, 1972,
Glass 1975.
[0214] The nonlinear differential equations are approximations of
the more realistic stochastic reaction framework. The stochastic
time evolution of this system is a Markov jump process where the
occurrence of each chemical reaction changes the concentration of
chemicals in discrete jumps as time moves forward. When certain
criteria are satisfied, an intermediate description between
nonlinear differential equations and the stochastic Markov process
emerges, i.e. nonlinear stochastic differential equations.
[0215] Implementation of the Mathematical Models in Software
[0216] The mathematical models can be implemented with existing
software packages. Basic information about the network, about the
reactions which occur in the network and the mathematical
frameworks which describe these reactions are input to the
programs.
[0217] The network information includes a list of chemicals and
their initial concentrations, a list of rate constants and their
values and a list of the reactions which take place in the network,
i.e. the reaction, the components and other chemicals involved in
the reaction, and the kinetic parameters involved in the reaction.
The list of reactions links up the components and other chemicals
present in the network to form the topology of the network.
[0218] Other reaction information which is input includes that
pertaining to reactions such as phosphorylation, dephosphorylation,
guanine nucleotide exchange, transport across the nuclear membrane,
transcription, translation and receptor binding. Each specific
reaction includes the chemicals that are involved in the reaction,
the stoichiometry of the reaction, i.e. the number of molecules
created or destroyed in the reaction, and the rate of the reaction.
The rate depends on the rate constants and the concentration of the
chemicals involved.
[0219] The mathematical frameworks, or reaction movers, are those
that can be used to evolve the state of the system forward in time.
They consist of differential equation dynamics and stochastic
dynamics movers. In the stochastic dynamics derived class, the
occurrence of a particular reaction is calculated in accordance
with the reaction rates entered. The concentrations of the
components are then changed to reflect the occurrence of the
reaction. The rates of each of the reactions is then recomputed.
Such a probabilistic time evolution of the biochemical network is
known as a continuous time Monte Carlo simulation. In the chemical
context it is known as the Gillespie algorithm (Gillespie 1976). In
the differential equation derived class, the differential equation
that describes the time rate of change for each component is
constructed from the kinetic forms and stoichiometry entered.
Various numerical integration routines, e.g. Runge-Kutta, are used
to solve for the new chemical concentrations as time moves forward.
Both the stochastic and nonlinear differential equation frameworks
output the concentration of all of the components in the network as
a function of time.
[0220] When time series data of protein or mRNA concentrations are
available for the particular system being studied, optimization
routines can be used to fit the rate constants. The values of the
rate constants are often not known and these optimization methods
can thus be used to make the model and the simulation of the model
more accurate. The system is first simulated at a particular set of
values for the rate constants. The resulting simulated time series
for a particular component is compared to an experimentally
measured concentration time series and a `penalty` or `cost` is
calculated as the sum of the squares of the differences between the
data and the simulated time series. The rate constants are then
perturbed away from the starting values and the simulation is
repeated and the cost recalculated. If the cost is lower after the
perturbation, the optimizer adopts the new set of rate constants
that resulted in the lower cost and a better fit to the data. The
perturbing or changing of the rate constants is sometimes performed
randomly and sometimes performed rationally, depending on the
optimization routine. The optimizer iterates the changing of the
rate constants, simulating the network, and evaluating the change
in the `cost` until the simulation nearly matches the data.
[0221] A measure of the predictive power of the mathematical model
is the robustness of the predictions obtained from the simulation
with optimized parameter values. Methods to accomplish this, known
as stochastic sensitivity analyses, are used. When a set of rate
constants are found that match the simulation to the data, the
parameter values are stochastically perturbed in the vicinity of
the minimum cost, and the system is simulated with this ensemble of
rate constant sets. If the output from the simulation does not vary
significantly within the ensemble of rate constant sets, the
prediction is robust and the predictive power of the model is high.
On the other hand, if the simulation varies significantly within
the ensemble of rate constant sets, the prediction is not robust
and the predictive power of the model is low.
EXAMPLE I
[0222] Description of a Network Comprising Two Genes
[0223] FIG. 5 is a DCL schematic representation of a network
comprising two genes, G.sub.A and G.sub.B. G.sub.A and G.sub.B are
transcribed independently from two separate promoters, P.sub.A and
P.sub.B, to produce mRNA A and mRNA B, respectively, which are then
translated to produce proteins A and B, respectively. Transcription
and translation are approximated as a single process. Protein A
inhibits the production of B. Proteins A and B together activate
the production of A. This is only physically plausible if the
operator DNA sequences in promoters P.sub.A and P.sub.B are
similar. P.sub.A.sup.total and P.sub.B.sup.total represent the
total number of promoter copies for genes A and B and is equal to
one for a single copy of the gene circuit. Promoter A.sub.7P.sub.A7
controls the production of protein A from gene A.sub.7G.sub.A.
Promoter B.sub.7P controls production of protein B from gene B.
Protein A represses production of protein B (indicated by -) while
protein A and protein B together activate the production of protein
A (indicated by +).
[0224] It has now been found that it is possible to describe the
state of the network, i.e., the concentrations of the gene
products, mRNA and proteins, as it evolves as a function of time,
by employing certain mathematical models.
[0225] The Mathematical Model
[0226] A deterministic model is established by deriving a set of
coupled nonlinear differential equations 2 x i t = f ( )
[0227] where index i labels a chemical species in the network, a
chemical species being a particular gene product, mRNA A, mRNA B,
protein A or protein B, or part of the gene itself, promoter A,
P.sub.A and the complexes that can be formed as a result of allowed
biochemical reactions [P.sub.A:A]. The following differential
equations for this system were integrated with a fourth-order
Runge-Kutta routine obtained from Numerical Recipes (Press et. al,
Numerical Recipes in C, 1992). 3 [ P A ] t = k uA ( [ P A : A ] + [
P A : B ] ) - k bA ( [ P A ] [ A ] + [ P A ] [ B ] ) ( 1 ) [ P A :
A ] t = k bA ( [ P A ] [ A ] - [ P A : A ] [ B ] ) + k uA ( [ P A :
A : B ] - [ P A : A ] ) ( 2 ) [ P A : B ] t = k bA ( [ P A ] [ B ]
- [ P A : B ] [ A ] ) + k uA ( [ P A : A : B ] - [ P A : B ] ) ( 3
) [ P A : A : B ] t = k bA ( [ P A : A ] [ B ] + [ P A : B ] ) - 2
k uA [ P A : A : B ] ( 4 ) [ mRNA A ] t = k tm [ P A : A : B ] - k
d 7 mRNA [ mRNA A ] [ A ] t = k tA [ mRNA A ] - k d [ A ] - k bA (
[ P A ] [ A ] + [ P A : B ] [ A ] ) + ( 5 ) k uA ( [ P A : A ] + [
P A : A : B ] ) - k bB ( [ P B ] [ A ] + k uB ( [ P B :A] ( 6 ) [ P
B ] t = k uB [ P B : A ] - k bB [ P B ] [ A ] ( 7 ) [ P B : A ] t =
k bB [ P B ] [ A ] - k uB [ P B : A ] ( 8 ) [ mRNA B ] t = k tm [ P
B ] - k d , 7 mRNA [ mRNA B ] [ B ] t = k tB [ mRNA B ] - k d [ B ]
- k b ( [ P A ] [ B ] + [ P A : A ] [ B ] ) + ( 9 ) k uA ( [ P A :
B ] + [ P A : A : B ] ) ( 10 )
[0228] The invention may be used to predict the behavior of the
network as represented by these ten differential equations. In one
method of the invention, the equations are solved analytically
(i.e. mathematically), if necessary, by making some approximations
and using tools from nonlinear dynamics or statistical physics. In
another method of invention, the equations are solved on a computer
by numerically integrating them and thereby simulating the
network's behavior as a function of time.
[0229] The Analytical Method Using Approximations
[0230] In general, the transcription rates k.sub.tA and k.sub.tB
for the two genes are different. The protein degradation rates are
assumed to be equal (k.sub.dA=k.sub.dB=k.sub.d) for simplicity.
This model assumes no multimerization. Such processes would not
qualitatively change the analysis or results that follow. The mass
balance for the promoters is expressed as follows:
P.sub.A.sup.total=[P.sub.A]+[P.sub.A:A:B] (11)
P.sub.B.sup.total=[P.sub.B]+[P.sub.B:A] (12)
[0231] These equations quantitatively state that the promoters can
be either free or bound to specific proteins. In equation (3) it is
assumed, for simplicity that [P.sub.A: A] and [P.sub.A:B] are much
smaller than [P.sub.A:A:B] and thus these terms do not appear. The
dynamics of free promoter concentrations are given by: 4 [ P A ] t
= k uA ( [ P A : A : B ] - k bA ( [ P A ] [ A ] [ B ] ( 13 ) [ P B
] t = k uB ( [ P B : A ] - k bB [ P B ] [ A ] ( 14 )
[0232] where k.sub.uA(k.sub.uB) and k.sub.bA (k.sub.bB) are the
unbinding and binding constants, respectively. The terms in these
equations correspond to the creation and destruction of free
promoter molecules, respectively. Assuming local equilibrium for
the promoter interactions, i.e. binding/unbinding happens at a much
faster rate than other processes in the cell, the equations are 5 [
P A ] t = 0 = k uA [ P A : A : B ] - k bA [ P A ] [ A ] [ B ] ( 15
)
[0233] for the steady state concentrations, leading to 6 K PA k bA
k uA = [ P A : A : B ] [ P A ] [ A ] [ B ] ( 16 )
[0234] Combining this result with the mass balance equations for PA
results in 7 [ P A : A : B ] = P A total K PA [ A ] [ B ] ( 1 + K
PA [ A ] [ B ] ) ( 17 )
[0235] A similar calculation can be made by rewriting [P.sub.B] in
terms of K.sub.PB and [A]. Assuming that transcription is very
fast, the dynamic equations for mRNA can be set equal to zero. With
these approximations, the following system of equations for the
dynamics of protein concentrations [A] and [B] emerge: 8 [ A ] t =
k tA [ P A : A : B ] - k d [ A ] ( 18 ) [ B ] t = k tB [ P B ] - k
d [ B ] ( 19 )
[0236] Inserting the expression derived for [PA:A:B] and [P.sub.B]
results in 9 [ A ] t = k tA P A total K PA [ A ] [ B ] 1 + K PA [ A
] [ B ] - k d [ A ] ( 20 ) [ B ] t = k tB P B total 1 K PB [ A ] +
1 - k d [ B ] ( 21 )
[0237] Thus, when a number of approximations are made (such as
lumping transcription and translation together as a single process,
assuming local equilibrium for promoter binding/unbinding, etc.),
the full system consisting of ten differential equations can be
reduced to a system of two differential equations. In this
framework of reduced dimensionality, the tools of nonlinear
dynamics can be employed to construct a phase portrait, analyze the
stability of fixed points, and hence predict the dynamics of the
system mathematically. The analysis of the two-dimensional system
that appears below gives rise to the phase portrait in FIG. 6. The
qualitative picture of the network dynamics provided by the phase
portrait is representative of the dynamics in the full
ten-dimensional system.
[0238] To simplify the analysis, the system is non-dimensionalized
by first dividing by k.sub.d so that time is now rescaled by
k.sub.d, i.e. t.fwdarw.k.sub.dt. Defining 10 A P A total k tA k d ,
B P A total k tB k d , A 1 / K PA k uA k bA , B 1 / K PB k uB k bB
,
[0239] equations (2) and (3) can be rewritten as 11 [ A ] t = A [ A
] [ B ] A + [ A ] [ B ] - [ A ] ( 22 ) [ B ] t - B B B + [ A ] - [
B ] ( 23 )
[0240] Setting these two equations equal to zero yields the two
nullclines displayed in the phase portrait in FIG. 6. There is a
fixed point at the intersection of these two nullclines, 12 ( B ( A
B - A ) A + B B , A + B B ) B + A
[0241] fixed point 1, and another fixed point at (0, aB), fixed
point 2.
[0242] The Jacobian J of a set of N differential equations is
defined as: 13 J = ( f 2 x 2 f 2 x 2 -- - -- - f 2 IN f 2 x 2 f 2 x
2 -- - -- - f 2 IN f N f 1 f N f 2 -- - f N f N ) . For the present
system , ( 24 ) J = ( A AB ( A + AB ) 2 - 1 A A A ( A + AB ) 2 B B
( B + A ) 2 - 1 ) ( 25 )
[0243] Analysis of the Jacobian matrix reveals that fixed point 1
is a node and fixed point 2 is a saddle point when
.alpha..sub.A.alpha..sub.B.- gtoreq..mu..sub.A. Fixed point 1 is
stable when: 14 A ( B + A ) A ( A + B B )
[0244] .gtoreq.2. This condition changes when the eigenvalues of
this Jacobian matrix become positive, indicating that fixed point 1
is now unstable and fixed point 2 is not stable. If fixed point 1
is the normal, healthy state of a cell and fixed point 2 is a
cancerous state of a cell, this stability condition identifies
which combination of parameters, and thus which combination of
genes, are most important in causing the transition or bifurcation
from the normal state to the cancer state. This in turn identifies
the genes which are putative targets for therapeutic agents for
treatment of the disease controlled by this particular network.
[0245] The ratio of transcription rates to degradation rates
(.alpha.'s) is normally greater than unity and the ratio of
unbinding constants to binding constants (.mu.'s) is typically much
less than unity so that fixed point 1 is a stable node for typical
parameter values. The analyses that follow take place in this
parameter range. The phase portrait in FIG. 6 indicates that the
system will flow to fixed point 1 given any non-zero initial values
of protein A. The system flows to fixed point 2 only in the absence
of protein A. The kinetic parameters used are set forth in Table
1.
1TABLE 1 Description of Parameters Symbol Value Units Cell Volume V
1.66 .times. 10.sup.-15 Litres Transcription Rate (mRNA.sub.A and
kt.sub.m 0.1 mRNA mlces./(DNA mlces. .times. sec.) mRNA.sub.B) mRNA
Degradation Rate kd, mRNA 1/30 1/sec. Translation Rate of A
kt.sub.A 1/3 protein mlces./(mRNA mlces. .times. sec.) Translation
Rate of B kt.sub.B 1/3 protein mlces./(mRNA mlces. .times. sec.)
Protein Degradation Rate kd 1/300 1/sec. Binding Rate of P.sub.A
kb.sub.A 1.25 .times. 10.sup.-4 1/sec. Binding Rate of P.sub.B
kb.sub.B 0.01 1/sec. Unbinding Rate of P.sub.A ka.sub.A 1/600
1/sec. Unbinding Rate of P.sub.B k.sub.Ub 1/600 1/sec.
[0246] Table 1 sets forth the parameters used in the differential
equation model of FIGS. 6 and 7. In FIGS. 7 and 8, both the
stochastic and differential equation systems are initialized with
50 protein A molecules and zero protein B molecules.
[0247] The mathematical equations describing the two gene network
are as follows: 15 [ PA ] t = uA k ( [ P A : A ] + [ P A : B ] ) -
bA k ( [ P A ] [ A ] + [ P A ] [ B ] ) ( 26 ) [ P A : A ] t = bA k
( [ P A ] [ A ] - [ P A : A ] [ B ] ) + uA k ( [ P A : A : B ] - [
P A : A ] ) ( 27 ) [ P A : B ] t = bA k ( [ P A ] [ B ] - [ P A : B
] [ A ] ) + uA k ( [ P A : A : B ] - [ P A : B ] ) ( 28 ) [ P A : A
: B ] t = bA k ( [ P A : A ] [ B ] + [ P A : B ] ) - uA 2 k [ P A :
A : B ] ( 29 ) [ mRNA A ] t = tm k [ P A : A : B ] - d , mRNA k [
mRNA A ] ( 30 ) [ A ] t = tA k [ mRNA A ] - d k [ A ] - bA k ( P A
] [ A ] + [ P A : B ] [ A ] ) + uA k ( P A : A ] + [ P A : A : B ]
) - bB k [ P 6 ] [ A ] + uB k [ P 6 : A ] ( 31 ) [ P 6 ] t = uB k [
P 6 : A ] - hB k [ P 6 ] [ A ] ( 32 ) [ P 6 : A ] t = bB k [ P 6 ]
[ A ] - uB k [ P 6 : A ] ( 33 ) [ mRNA 6 ] t = tm k [ P 6 ] - d ,
mRNA k [ mRN A 6 ] ( 34 ) [ B ] t = tB k [ mRN A 6 ] - d k [ B ] -
b k ( [ P A ] [ B ] + [ P A : A ] [ B ] ) + uA k ( P A : B ] + ( [
P A : A : B ] ) ( 35 )
[0248] The set of equations (non-approximated) that quantitatively
describe the time evolution of the concentration of gene products
in this system are numerically integrated. The results of this
numerical integration simulate the behavior of the two gene network
and results in the plots shown in FIGS. 6 through 8. The time
course of these plots corresponds to a particular biological state.
For example, one particular time course can correspond to the
normal progression of a cell through the cell cycle while another
time series can correspond to the unregulated cell growth
characteristic of cancer. By perturbing particular genes or
proteins in the circuit, the time series of a cancerous cell can be
changed into the time series of a healthy cell, thereby identifying
sets of genes and proteins as putative targets for therapeutic
agents.
[0249] FIGS. 7 and 8 compare the output of the stochastic model and
the differential equation model for the two gene network. FIG. 7
displays the time evolution of the differential equation model for
a single copy of the gene circuit. The system quickly flows to the
stable fixed point 1 as expected. In the differential equation
model, the system would never reach the "extinction" fixed point
unless the system started with zero A molecules.
EXAMPLE II
[0250] Description of the Wnt .beta. Catenin Pathway
[0251] FIG. 10 contains a graphical representation of the Wnt
.beta.-catenin pathway indicating the role of Axin, APC, and GSK3
in phosphorylating .beta.-catenin and leading to its degradation.
FIG. 10 was created as well using Diagrammatic Cell Language, which
was discussed above in connection with FIG. 3.
[0252] In FIG. 10, there are two broad horizontal lines, CM and NM.
The upper broad line CM represents schematically the cell membrane;
that is, the outer membrane of the cell, and the lower broad line
NM represents the nuclear membrane of the cell. Elements below the
line NM are in the nucleus, and elements above the line CM are
outside of the cell.
[0253] Referring again to FIG. 10, Wnt signaling is induced by
secreted Wnt proteins that bind to a class of seven-pass
transmembrane receptors encoded by the frizzled genes. Activation
of the frizzled receptor leads to the phosphorylation of disheveled
(Dsh) through an unknown mechanism. Activated disheveled inhibits
the phosphorylation of .beta.-catenin by glycogen synthase kinase
3.beta. (GSK 3 .beta.). Unphosphorylated .beta.-catenin escapes
detection by .beta.-TrCP which triggers the ubiquitination of
.beta.-catenin and its degradation in the proteasomes. Stabilized
.beta.-catenin, as a result of Wnt signaling, enters the nucleus
where it interacts with TCF/LEF1 transcription factors leading to
the transcription of Wnt target genes such as CyclinD1 and
c-Myc.
[0254] In the absence of Wnt, .beta.-catenin phosphorylation by
GSK3.beta. occurs in a multiprotein complex containing the
scaffolding protein Axin, as well as GSK3.beta. and the APC tumor
suppressor. In the multiprotein complex, .beta.-catenin is
efficiently phosphorylated and then is earmarked for degradation by
.beta.-TrCP. Stabilized .beta.-catenin is common to most colon
cancers, where mutations in APC, Axin, and .beta.-catenin itself
are known to interfere with its effective ubiquitination and
consequently its degradation. Accumulation of .beta.-catenin leads
to the activation of the Wnt target genes such as CyclinD1 and
c-Myc, both of which are intimately involved in cell cycle control
and the progression of cancer. Nuclear .beta.-catenin also targets
.beta.-TrCP increasing its levels and creating a negative feedback
loop in the system.
[0255] Common mutations found in colon and other cancers usually
effect the NH.sub.2-terminal phosphorylation of .beta.-catenin, the
binding of APC to Axin, and or mutations in Axin that prevent
.beta.-catenin from binding to Axin. These mutations can be
represented in the simulation by deleting reactions and setting the
rate constants that correspond to these reactions equal to
zero.
[0256] FIG. 10 depicts a subnetwork that represents the components
involved in Wnt signaling in addition to side pathways responsible
for 1-catenin degradation. These include the Axin degradation
machinery and 1-catenin transcription of target genes c-Myc and
.beta.-TrCP. The representation includes notations depicting all of
the components, chemical forms of the components, and reactions
involved in the network in a complete yet compact manner. It is
directly translatable to various mathematical descriptions.
[0257] Simulation of the Network
[0258] All of the chemical species in the system are listed. Each
chemical species is or may be involved in a reaction. The time
course of its quantity or concentration is simulated. In this
example, the components include: Axin, .beta.-catenin, APC, GSK3,
.beta.-TrCP, HDAC, Groucho, c-Myc gene, c-Myc mRNA, .beta.-TrCP
gene, .beta.-TrCP mRNA, and an unknown intermediary protein that
facilitates the enhancement of .beta.-TrCP by nuclear
.beta.-catenin. Each of these components can exist in an alternate
form depending on the species with which it interacts. For example,
.beta.-catenin can be phosphorylated directly by GSK3 forming
.beta.-catenin phosphorylated. It can also bind to Axin to form a
.beta.-catenin:Axin complex. There are a total of 70 components and
chemical species in this exemplary simulation. They are listed in
Table 2 below.
2TABLE 2 Components and Chemical Species in the Wnt b-catenin
Network <APC> <APCAxin> <APCAxinG>
<APCAxinp> <APCAxinpG> <APCB> <APCBAxin>
<APCBAxinG> <APCBAxinp> <APCBAxinpG>
<APCBP> <APCBpAxin> <APCBpAxinG>
<APCBpAxinp> <APCBpAxinpG> <APCp>
<APCpAxin> <APCpAxinG> <APCPAXinp>
<APCpAxinpG> <APCpB> <APCpBAxin>
<APCpBAxinG> <APCpBAxinp> <APCpBAxinpG>
<APCpBp> <APCpBpAxin> <APCPBpAxinG>
<APCpBpAxinp> <APCpBpAxinpG> <AXin> <AxinG>
<Axinp> <AxinpG> <B> <BAPC> <SAPCp>
<BAxin> <BAxinG> <BAxinp> <BAxinpG>
<BBTCFcKycGene> <BBOACGroucho> <Bnuclear>
<BP> <SpAPC> <BpAPCp> <BpAxin>
<BpAxinG> <SpAxinp> <BpAxinpG>
<BPBPTCFcMycGene> <BpBTCFcMycGene>
<BpHDACGroucho> <Bpnuclear> <BpTCFcMycGene>
<BTCFcKycGene> <bTrCP> <bTrCPBpUbUb>
<bTrCPGene> <bTrCPmRNA> <cMyCMRNA> <G>
<NDACGroucho> <HDACGrouchoTCFcMycG ene>
<Intermediary> <Source> <SourceB>
<TCFcMycGene> <zero>
[0259] The interactions, i.e. reaction steps or binding
interactions, between the components are listed in Table 3,
below.
3TABLE 3 List of Reactions, kinetic forms and kinetic parameters
Stochlometry of the Reaction: Chemicals entering the reaction and
Chemicals emerging from the reaction Reaction Type Kinetic Paramter
1 .fwdarw. (1 Bp) MichaelisMentenKF 2 3 .fwdarw. (1 APCp)
MichaelisMentenKF 4 5 .fwdarw. (1 AxinG) SimpleKF (kbAxintoG)
(-AxinG) .fwdarw. 6 SimpleKF (kuAxintoG) (-AxinG) .fwdarw. (1
AxinpG) SimpleKF (kbG) 7 .fwdarw. (1 AxinpG) SimpleKF (kbAxintoG)
(-1 AxinpG) .fwdarw. 8 SimpleKF (kuAxintoG) 9 .fwdarw. (1 BAxin)
SimpleKF (kbBtoAxin) (-1 BAxin) .fwdarw. 10 SimpleKF (kuBtoAxin) 11
.fwdarw. (1 BpAxin) SimpleKF (kbBtoAxin) (-1 BpAxin) .fwdarw. 12
SimpleKF (kuBtoAxin) 13 .fwdarw. (1 BAxinp) SimpleKF (kbBtoAxinp)
(-1 BAxinp) .fwdarw. 14 SimpleKF (kuBtoAxin) 15 .fwdarw. (1
BpAxinp) SimpleKF (kbBtoAxinp) (-1 Axinp) .fwdarw. 16 SimpleKF
(kuBtoAxinp) 17 .fwdarw. (1 APCAxin) SimpleKF (kbAPCtoAxin) (-1
APCAxin) .fwdarw. 18 SimpleKF (kuAPCtoAxin) 19 .fwdarw. (1
APCpAxin) SimpleKF (kbAPCtoAxin) (-1 APCpAxin) .fwdarw. 20 SimpleKF
(kuAPCtoAxin) 21 .fwdarw. (1 APCAxinp) SimpleKF (kbAPCtoAxin) (-1
APCAxinp) .fwdarw. 22 SimpleKF (kuAPCAxin) 23 .fwdarw. (1 APCpAxin)
SimpleKF (kbAPCtoAxin) (-1 APCAxinp) .fwdarw. 24 SimpleKF
(kuAPCtoAxin) 25 .fwdarw. (1 BAPC) SimpleKF (kbBtoAPC (-1 BAPC)
.fwdarw. 26 SimpleKF (kuBtoAPC) 27 .fwdarw. (1 BpAPC) SimpleKF
(kbBtoAPC) (-1 BpAPC) .fwdarw. 28 SimpleKF (kuBtoAPC) 29 .fwdarw.
(1 BAPCp) SimpleKF (kbBtoAPCp) (-1 BAPCp) .fwdarw. 30 SimpleKF
(kuBtoAPCp) 31 .fwdarw. (1 BpAPCp) SimpleKF (kbBtoAPCp) (-1 APCp)
.fwdarw. 32 SimpleKF (kuBtoAPCp) 33 .fwdarw. (1 BAxinG) SimpleKF
(kbAxintoG) (-1 BAxinG) .fwdarw. 34 SimpleKF (kuAxintoG) 35
.fwdarw. (1 BpAxinG) SimpleKF (kbBtoAxin) (-1 BpAxinG) .fwdarw. 36
SimpleKF (kuBtoAxin) 37 .fwdarw. (1 BpAxinG) SimpleKF (kbAxintoG)
(-1 BpAxinG) .fwdarw. 38 SimpleKF (kuAxintoG) 39 .fwdarw. (1
BAxinpG) SimpleKF (kbBtoAxinp) (-1 AxinpG) .fwdarw. 40 SimpleKF
(kuBtoAxinp) 41 .fwdarw. (1 BAxinpG) SimpleKF (kbAxintoG) (-1
BAxinpG) .fwdarw. 42 SimpleKF (kuAxintoG) 43 .fwdarw. (1 BAxinpG)
SimpleKF (kbBtoAxinp) (-1 AxinpG) .fwdarw. 44 SimpleKF (kuBtoAxinp)
45 .fwdarw. (1 BpAxinpG) SimpleKF (kbAxintoG) (-1 AxinpG) .fwdarw.
46 SimpleKF (kuAxintoG) 47 .fwdarw. (1 BpAxinG) SimpleKF
(kdBtoAxinp) (-1 BpAxinpG) .fwdarw. 48 SimpleKF (kuBtoAxinp) (-1
BAxinG) .fwdarw. (1 BpAxinG) SimpleKF (kbGWAxin) (-1 BAxinG)
.fwdarw. (1 BAxinpG) SimpleKF (kbGWAxin) (-1 BpAxinG) .fwdarw. (1
BpAxinG) SimpleKF (kbGWAxin) (-1 BAxinpG) .fwdarw. (1 BpAxinG)
SimpleKF (kbGWAxin) 49 .fwdarw. (1 APCBAxin) SimpleKF
(kbAPCtoAkin)) (-1 APCBaxin) .fwdarw. 50 SimpleKF (kuAPCtoAxin) 51
.fwdarw. (1 APCBaxin) SimpleKF (kbAPCtoAxin) (-1 APCBaxin) .fwdarw.
52 SimpleKF (kuAPCtoAxin) 53 .fwdarw. (1 APCBpAxin) SimpleKF
(kbAPCtoAxin) (-1 APCBpAxin) .fwdarw. 54 SimpleKF (kuAPCtoAxin) 55
.fwdarw. (1 APCBpAxin) SimpleKF (kbAPCtoAxin) (-1 APCBpAxin)
.fwdarw. 56 SimpleKF (kuAPCtoAxin) 57 .fwdarw. (1 APCpBAxin)
SimpleKF (kbAPCtoAxin) (-1 APCpBAxin) .fwdarw. 58 SimpleKF
(kuAPCtoAxin) 59 .fwdarw. (1 APCpBAxin) SimpleKF (kuAPCptoB) (-1
APCB) .fwdarw. 60 SimpleKF (kuAPCptoB)
[0260] These interactions are translated into a mathematical
kinetic form. For example, the binding of Axin to .beta.-catenin
has the following mathematical form:
[Axin][.beta.-catenin]k.sub.b
[0261] where the quantities in brackets represent the
concentrations of the two proteins. Not all binding reactions need
be represented in an equivalent kinetic form. For example, two
receptors binding on the membrane surface may be better represented
by the mathematical kinetic form:
[ReceptorI].sup.a[ReceptorII].sup..beta.k.sub.b
[0262] where .alpha. and .beta. are constants greater than 1. This
form may better represent the kinetics of proteins interacting in a
restricted geometry. Those skilled in the art can derive and
express the appropriate kinetic form depending on the geometry, the
reactants and the nature of the reaction.
[0263] Table 3 lists all of the reactions incorporated into the
simulation together with the kinetic form used to represent the
reaction and the corresponding kinetic rate constants and their
values.
[0264] In the stochastic embodiments of the invention, each
reaction represents a probability of a reaction occurring. In the
deterministic embodiments each reaction represents a term in the
differential equation representing the time rate of change of the
chemical species. The list of differential equations is set forth
in Table 4 below.
[0265] Simulations
[0266] The initial values of the kinetic parameters are chosen from
the literature by incorporating time scale and expression
information. For example, it is known that GSK3 phosphorylates Axin
on a time scale of about 30 minutes, and hence a rate constant is
chosen to reflect that time scale.
[0267] Set forth below are deterministic and stochastic solutions
of what is considered the "normal state" of the cell. Low levels of
.beta.-catenin and of .beta.-catenin target genes such as c-Myc
characterize the normal state. In the deterministic solution, the
time series profile characterizes the "normal" state where the Axin
degradation machinery keeps the levels of .beta.-catenin low and
consequently limits the levels of downstream targets such as the
proto-oncogene c-Myc. This is shown in FIG. 11.
[0268] Perturbations can be introduced to determine the relevant
targets in the network and the effect they have on perturbing the
network. For example, the binding rate of Axin to APC can be set to
zero, thereby simulating the effects of a mutation in APC that
prevents its binding to Axin. FIG. 12 depicts a time series profile
which characterizes the "cancerous" state, for example, where a
mutation in APC prevents Axin from effectively degrading
.beta.-catenin. This results in the up-regulation of c-Myc as well
as higher levels of .beta.-TrCP. In this case, the level of
.beta.-catenin rises in the cytoplasm and the nucleus and
consequently c-Myc transcription rises as well.
EXAMPLE III
[0269] One or more components of a cell can be identified as
putative targets for interaction with one or more agents within the
simulation. This is achieved by perturbing the simulated network by
deleting one or more components thereof, changing the concentration
of one or more components thereof or modifying one or more of the
mathematical equations representing interrelationships between two
or more of said components. Alternatively, the concentrations of
one or more of the several proteins and genes in the biochemical
network are selectively perturbed to identify which ones of said
proteins or genes cause a change in the time course of the
concentration of a gene or protein implicated in a disease state of
the cell.
[0270] Deleting One or More Components in the Network
[0271] The APC protein is deleted by removing the protein from the
set of equations in Example II and removing all of the chemical
species formed and reactions that take place as a result of
interactions with APC. The effect on the state of the cell is to
raise the levels of .beta.-catenin so that it continually activates
downstream targets such as .beta.-TrCP and c-Myc. This can be seen
by comparing FIG. 13 with FIG. 11. APC is thus identified as an
important component of the cell which is implanted in a disease
state of the cell. When APC is "knocked" out it leads to a high
level of .beta.-catenin which can cause the development and
progression of colon cancer. In the event that a mutation in APC
prevents its interaction with the components in the cell,
therapeutics can be sought to rectify this condition.
[0272] HDAC, a protein that sequesters nuclear .beta.-catenin and
represses the c-Myc gene, is deleted by removing HDAC from the set
of equations set forth in Example II and thereby removing all of
the chemical species that are formed and reactions that take place
as a result of interactions with HDAC. The effect on the state of
the cell, as simulated, is an increase in the levels of c-Myc. This
is shown in FIG. 14. HDAC is thus identified as an important target
because when it is "knocked" out, high levels of c-Myc develop and
this can lead to the development and progression of colon cancer.
In the event that a mutation in HDAC prevents its interaction with
the components in the cell, therapeutics can be sought to rectify
this condition.
[0273] Changing the Concentrations of One or More the Components in
the Network
[0274] Starting from the simulated cancerous state depicted in FIG.
12, the concentration of Axin is increased significantly above its
normal levels. This results in the reduction of .beta.-catenin
levels. This is shown in FIG. 15. This identifies a cellular
component, Axin, that can cause a reduction in the time course of
the concentration of .beta.-catenin.
[0275] The levels of HDAC are then increased in the simulated cell
to which Axin has been introduced. This lowers the concentrations
of nuclear .beta.-catenin and c-Myc. Both levels are lowered
recreating a profile that corresponds to a "normal" cellular state.
This is shown in FIG. 16. This identifies HDAC as an important
component for control of a disease state.
[0276] Starting from the disease state of FIG. 12, the
concentrations of Axin and GSK3 are perturbed by increasing their
levels significantly above normal. The concentration of Axin is
perturbed less than above so that .beta.-catenin levels fall, but
not as much. GSK3 concentration is then increased to further reduce
.beta.-catenin levels. The levels of .beta.-catenin approach that
of the normal state. This is shown in FIG. 17. This identifies Axin
and GSK3 as two components which affect the time course of
.beta.-catenin.
[0277] A further series of perturbations are made, each of the
perturbations changing the concentration of a protein or gene in
the network to a perturbed value to determine whether that protein
or gene is implicated in causing a change in the time course of the
concentration of a gene or protein implicated in a disease state of
the cell. Starting from the disease state of FIG. 12, the
concentrations of Axin and GSK3 are perturbed by increasing their
levels significantly above normal. This is shown in FIGS. 18 and
19, respectively. The system is perturbed again by raising the
levels of HDAC. This is shown in FIG. 20. Upon each perturbation,
the levels of .beta.-catenin are reduced in the cytoplasm and then
in the nucleus resulting in reduced levels of c-Myc mRNA. This
identifies Axin, GSK3, and HDAC in varying degrees, as causing a
change in the time course of the concentration of the gene or
protein implicated in the disease state.
[0278] Starting from the normal state of FIG. 11, the sequence of
the perturbations described above is repeated, but the
concentration of each of the respective components is sequentially
set to zero. This raises the cytoplasmic levels of .beta.-catenin
for a short duration. Then it raises it significantly over the
entire time course. Finally, the levels of nuclear .beta.-catenin
are increased further as a result of the final perturbation. This
is shown in FIGS. 21, 22 and 23. This identifies Axin, GSK3, and
HDAC as components whose interactions have a significant effect on
the state of the cell.
[0279] Modifying the Mathematical Equations
[0280] The mathematical equations in Example II are modified by
adding a new component to facilitate the binding of Axin to
.beta.-catenin. The reaction term that represents the binding of
Axin to .beta.-catenin is changed from
[Axin][.beta.-catenin]k.sub.b
to
[Axin][.beta.-catenin][Facilitator]k.sub.bwithFaciltator
[0281] where the binding with the Facilitator molecule is greater
than without the Facilitator. This perturbation is made to the
simulated "cancerous" state of FIG. 12 and is shown in FIG. 23.
This identifies the Facilitator as an important putative
therapeutic for changing the binding of Axin to .beta.-catenin and
thereby changing the condition of the cell from the disease state
to the healthy state.
[0282] Starting from the disease state of FIG. 12, the binding rate
of Axin to .beta.-catenin is increased. This perturbation allows
Axin to bind to .beta.-catenin more quickly and thus enable its
phosphorylation and degradation without APC. This is shown in FIG.
24. This identifies Axin as a component of the cell that changes
the time course of the disease state.
[0283] Starting from the disease state of FIG. 12, the binding rate
of Axin to .beta.-catenin is increased slightly. Then the binding
rate of .beta.-catenin to the c-Myc TCF bound gene is decreased.
Then the binding rate of GSK3 to Axin is increased. This results in
a decrease in .beta.-catenin levels and c-Myc transcription levels.
The effect on the time series profile of each successive
perturbation is shown in FIGS. 25, 26 and 27. This identifies Axin,
the TCF bound c-Myc gene, and GSK3 as important targets for
intervention.
[0284] Starting from the normal state of FIG. 11, the parameter for
the binding of Axin to GSK3 is set to zero and then the parameter
for the unbinding of .beta.-catenin to the c-Myc gene is set to
zero. The first perturbation causes a significant increase in
cytoplasmic and nuclear .beta.-catenin. The second perturbation
increases c-Myc mRNA levels immediately. This is shown in FIGS. 28
and 29. This identifies GSK3 and nuclear .beta.-catenin with the
c-Myc gene as important targets for affecting the time series
profile of the disease state.
[0285] Starting from the disease state of FIG. 12, the parameters
in the system are systematically perturbed until the profile of the
time series expression looks "normal." The simulation is run at the
starting kinetic and concentration values of the disease state and
then a computer code is executed to systematically vary one or more
kinetic parameter and concentration value from 0 to some maximum
number until the desired time series profile is reached. A
criterion is introduced to cease the perturbation when the time
series matches the desired output, in this case, that of a "normal"
profile similar to FIG. 11. The systematic changes are shown in
FIG. 31 and consist of a final change where the binding rate of
B-catenin to Axin is increased, the binding rate of B-catenin to
c-Myc TCF bound gene is decreased, and the binding rate of GSK3 to
Axin is increased.
[0286] Colon Cancer Model
[0287] A colon cell containing over 200 genes and proteins, over
800 states, and 900 parameters has been simulated. The colon cell
simulation incorporates the following networks: Ras MAPK, PI3K/AKT
signaling, Wnt signaling through beta-catenin, TGF-beta signaling,
TNF signaling, JAK/STAT pathways, NFkB, Fas and TNF signaling
through the caspases, and the highly connected p53 node. FIGS. 31
and 32 show the modular description whereby modular we mean a
simplification of the model into basic elements to clearly see
gross connections between components, major feedback loops, and
cross talk between the modules. Each module contains many reaction
steps. The lines extending from the modules indicate interactions
between the modules. Exhibit A contains the differential equations
for each module, the chemical species or states in each module, as
well as a list of the initial concentrations and kinetic parameters
used in the simulation.
[0288] The eight modules comprising FIG. 32 are, for convenience,
designated with a quadrant and a right/left deisgnation in order to
easily orient a given individual module with the overall modular
description of FIG. 32. As well, the interconnections between the
various modules are noted on each module, and are summarized in the
following table, where for convenience they are designated as
"lines", actually referring to biological interconnections.
Table of Interconnections Among the Octants of the Modular Cell
Diagram
[0289] The following table represents the interconnections among
the octants of the large modular cell diagram. Each interconnecting
line is numbered in a clockwise manner depending on the point where
the line exits a particular octant in the large diagram. Each
octant corresponds to the following biological system, subsystem or
pathway: NW-L=Ras MAPK; NW-R=PI3K/AKT; NE-L=Wat B-catenin;
NE-R=TGF-B; SE-R=p53; SE-L=apoptosis; SW-R=G1-S,G2-M; and
SW-L=JAK/STAT.
4 Destination Corresponding Line Original Site Octant Destination
Site Line NW-L1 [node] NW-R ErbB-R NW-R6 NW-L2 [node] NW-R RAS:GTP
NW-R5 NW-L3 ErK.sub.Nucl NE-L Erk* NE-L1 NW-L4 p90RSK SE-L P90RSK*
SE-L10 NW-L5 cycD NE-L cyclinD NE-L3 SW-R cycD SW-R7 NW-L6 Cip SW-R
Cip SW-R6 NW-L7 (PKB:PIP3).sub.2 NW-R (PKB:PIP3).sub.2 NW-R4 NW-R1
active TNF- SE-L [node] SE-L2 R/TRADD/ TRAF/RIP NW-R2
(PKB:PIP3).sub.2 SE-L (PKB:PIP3).sub.2 SE-L6 NW-R3 NFkB.sub.NuCl
SE-L NFkB.sub.NuCl SE-L1 NW-R4 (PKB:PIP3).sub.2 NW-L
(PKB:PIP3).sub.2 NW-L7 NW-R5 RAS:GTP NW-L [node] NW-L2 NW-R6 active
NW-L [node] NW-L1 ErbB-R NE-L1 Erk* NW-L ErK.sub.Nucl NW-L3 NE-L2
myc SE-R c-myc SE-R3 NE-L3 cyclin D NW-L cycD NW-L5 SW-R cycD SW-R7
NE-R1 c-jun SE-L jun* SE-L5 NE-R2 Jnk* SE-L JNK.sub.Nuc* SE-L4
NE-R3 Ink SW-R Ink SW-R1 SE-R1 p53* SW-R p53 SW-R4 SE-R2 E2F SW-R
E2F SW-R2 SE-R3 c-myc NE-L myc NE-L2 SE-R4 p53* SE-L p53* SE-L3
SE-L1 NFkB.sub.Nucl NW-R NFkB.sub.Nucl NW-R3 SE-L2 [node] NW-R
active TNF- NW-R1 R/TRADD/ TRAF/RIP SE-L3 p53* SE-R p53* SE-R4
SE-L4 JNK.sub.Nuc* NE-R Jnk* NE-R2 SE-L5 jun* NE-R c-jun NE-R1
SE-L6 (PKB:PIP3).sub.2 NW-R (PKB:PIP3).sub.2 NW-R2 SE-L7
Bcl-X.sub.L SW-L Bcl-x.sub.L SW-L3 SE-L8 FASL SW-L FASL SW-L2 SE-L9
14-3-3 sigma SW-R cycB SW-R3 SE-L10 P90RSK* NW-L P90RSK* NW-L4
SW-R1 Ink NE-R Ink NE-R3 SW-R2 E2F SE-R E2F SE-R2 SW-R3 cycB SE-L
14-3-3 sigma SE-L9 SW-R4 p53 SE-R p53* SE-R1 SW-R5 cycD SW-L cycD
SW-L1 SW-R6 Cip NW-L Cip NW-L6 SW-R7 cycD NW-L cycD NW-L5 NE-L
cyclinD NE-L3 SW-L1 cycD SW-R cycD SW-R5 SW-L2 FASL SE-L FASL SE-L8
SW-L3 Bcl-x.sub.L SE-L Bcl-X.sub.L SE-L7
[0290] Each module contains components that are important for colon
cancer progression and the mammalian cell cycle in general. The Ras
MAPK module of FIG. 32(a) contains important growth factors such as
EGF, TGF-alpha, and amphiregulin that activate the Erb family of
receptors. Once activated these receptors activate a cascade of
proteins called Ras, Mek, and Erk. Active Erk plays an important
role in turning on genes that are responsible the mammalian cell
cycle, genes such as cyclinD which initiate the transition from G1
to S phase. Erk even intiates further cellular division by turning
on the same growth factors that lead to its activation in an
autocrine manner. Often times, cancer will have a mutations in Ras
that inhibits its hydrolysis or conversion from its active GTP
bound form to its inactive GDP bound form, thereby promoting
activation of Erk and proliferation.
[0291] The Ras MAPK module also interacts with the P13K/AKT module
of FIG. 32(b). The same growth factors that lead to Erk activation
can also activate the survival factor AKT (sometimes referred to as
PKB). AKT can also be activated by a set of growth factors known as
Insulin, IGF-1, and IGF-2 as shown in the module description. Once
activated by growth factor signals, AKT induces the nuclear
translocation of NFkappaB. NFkappaB turns on a set of genes that
up-regulate proteins labeled as survival factors such as Bcl2,
Bcl-xL, Flips, and LIPs shown in the apoptosis module of FIG.
32(g). These proteins inhibit apoptosis or programmed cell death on
many levels. Thus a healthy dividing cell will signal to promote
cellular division and to inhibit apoptosis via upregulation of
these survival factors.
[0292] The Wnt beta-catenin module of FIG. 32(c) is another module
that signals to promote cellular division. This pathway usually
contains a mutation in Beta-catenin that leads to high levels of
the protein in the cell. Normally, the pathway is inactive and
Beta-catenin levels are low unless the cell is stimulated with the
Wnt ligand. Beta-catenin also acts as a transcription factor
turning on CylinD and c-Myc both of which lead to the G1-S
transition. Excess levels can thus lead to proliferation and
uncontrolled cellular division. In addition to promoting cellular
division, cancer cells will inhibit the process of differentiation.
A colon cancer cell achieves this by mutating the SMADS in the
TGF-beta pathway of FIG. 32(d). These are again transcription
factors activated by the TGF-beta ligand that promote the
transcription of genes like p21 and p16 which halt the cell cycle
signaling that it is time for the cell to differentiate.
[0293] The pathways described above signal to the G1-S and G2-M
modules of FIG. 32(f) which control the core cell cycle machinery.
Here CyclinD-CDK2/4 complexes are activated by growth factors to in
turn activate CylinE-CDK2 complexes which initiates DNA synthesis
or S phase. Upon the complition of S phase, CylinA-CDK2 complexes
are activates to induce chromosomal replication. The completion of
this process is marked by activation of CylinB-CDK1 complexes that
induce the onset of M phase or mitosis terminating in the cell
successfully replicating its DNA and dividing into two daughter
cells.
[0294] Often times, though, the cell makes errors in replicating
its DNA or in other phases of the cell cycle. In this case, the p53
transcription factor of FIG. 32(h) is activated to up-regulate
genes that halt the cell cycle and or genes that promote apoptosis
(e.g. Bax and Bad) should the cell not correct its defects in time.
p53 is a very commonly mutated gene in colon cancer and thus rather
than repairing its DNA or undergoing programmed cell death when the
DNA is damaged or mistakenly replicated, the cell can continue to
divide. Other signals that effect the state of the cell are
received via the JAK/STAT pathway where cytokines activate
components like p38 and JNK shown as JAK/STAT in FIG. 32(e). These
again, can up-regulate transcription factors that promote cellular
death (amongst other signals that even compete with apoptosis) via
up-regualtion of death inducing cytokines such as FASL when the
cell is stressed.
[0295] The apoptosis signals converge onto FIG. 32(g). Here
apoptosis can be trigerred via activation of death receptors such
as FASR and TNFR through the ligands TNF-alpha and FASL. These
receptors activate caspase 8 leading to cleavage of Bid which
induces the oligomerization of Bad. Bad can disrupt the
mitochondrial integrity releasing cytocrome c and activating
caspase 9 which in turn activates the executioner caspase 3. The
executioner caspases cleave various proteins in the cell and induce
programmed cell death. Caspase 8 can also directly activate caspase
3. Many colon cells require both direct caspase 3 activation and
disruption of the mitochondrial integrity leading to cytocrome c
release to fully activate the executioner caspases and subsequently
apoptosis.
[0296] By scaling the model to include the networks responsible for
the physiological process of the G1-S transition, S phase, G2-M
transition, and apoptosis we can use the model to predict the
various physiological states of the cell. In addition, by including
the major signal transduction pathways that contain key mutations
common to colon cancer we can simulate and understand the various
stages of colon cancer as given by a series of point mutations. The
simulation can be used to model each stage of the disease
progressing from a normal state with no muations to carcinoma where
multiple mutations have accumulated. Therapeutic strategies can be
suggested for each stage of the disease. In addition mutations
specific to an individual can be inputted devise individual
targeted therapies. Data from that individual on the DNA level, RNA
level, and protein level can be incorporated and optimized to this
core skeletal model to generate an optimal therapeutic strategy.
Technologies are becoming available and widely used to make such
patient specific data available.
[0297] The network is simulated as a whole. All of the differential
equations are solved simultaneously. One can perturb any of the
cellular components in the simulation to predict cellular outcome
and understand the cross talk and feedback loops between the
various modules. In the simulation, cellular mechanisms are
represented on multiple levels, including receptor activation,
degradation, endocytosis, signal transduction cascades, transport
within compartments, transcriptional control of gene expression
networks, and protein translation and degradation mechanisms.
[0298] Optimization of the Model: Determining and Constraining
Parameter Values
[0299] Time course measurements of the protein and mRNA levels were
incorporated into the model to constrain parameter values using
optimization algorithms. These algorithms are incorporated in
software, e.g. the Implementation software which simulates HCT116,
SW480, and Caco-2 colorectal cell lines. FIGS. 33(a)-(d), show time
course profiles of HCT116 cells under 20 ng of EGF stimulation. The
figures plot phosophorylated MEK, ErK, AKT, and RAF and simulation
data that has been optimized to fit the model. The solid lines show
the simulation output and the dots are the measured data from
Caco-2 cells stimulated with 20 ng EGF.
[0300] Optimization of the model is carried out as follows. When
the model is entered into the modeling software some parameter
values are known from the literature, e.g. kinetic binding rate
constants, phosphorylation rate constants, . . . etc. Some however,
are not known, and putative values for those rate constants must be
entered. Rate constan values can be gathered from literature
sources that have measured their values or by estimating their
value from what is known in the literature or otherwise on the
activation or deactivation of a particular component or analgious
biological component if that information is not available. These
starting values may produce simulation outputs that do not
necessarily match up with experimental time course measurements of
the expression levels of the actual components. The expression
levels are the total protein levels, levels of modified forms of
the protein (e.g. phosophorylated, cleaved,), and or RNA levels
using a multitude of experimental methods.
[0301] FIGS. 33(a)-(d) contain the data points for the
phosphoryatled forms of AKT, MEK, and ERK. These data points are
fed into the simulation and the resulting simulated time series for
a particular chemical is compared to an experimentally measured
concentration time series. A `penalty` or `cost` is calculated as
the square of the difference between the data and the simulated
time series. The cost function for each experiment,
CF.sup.i(kb.sub.1,ku.sub.2,kp.sub.1,kM.sub.2, . . . ) is a function
of all of the parameters in the model and is defined as 16 CF i (
kb 1 , ku 2 , kp 1 , kM 2 , ) = j ( Exp_data j i - Sim_data j i ) 2
j i 2 ,
[0302] where Exp_data.sub.j.sup.i, Sim_data.sub.j.sup.i, and
.sigma..sub.j.sup.i are the experimental data point, the simulation
data point, and the error on each data point for an experiment i,
respectively. Other cost functions exist which take into account
error in time measurements. The sum over j is taken over all of the
data points collected from the experiment. To incorporate the cost
from every experiment, e.g. a different experiment would consist of
stimulating with another growth factor or a different dose of
growth factor or inhibiting a particular component in the network .
. . etc., the global cost function, CF(kb.sub.1,ku.sub.2,kp.sub.1,
kM.sub.2, . . . ), which is a sum over all experiments i is
calculated as 17 CF ( kb 1 , ku 2 , kp 1 , kM 2 , ) = i CF i ( kb 1
, ku 2 , kp 1 , kM 2 , ) .
[0303] Each simulation data point is computed with the conditions
specific to that experiment. For example, an EGF experiment with
five different levels of EGF would be simulated under each of those
conditions and similarly for other treatments and conditions. The
goal is to find the parameter values that minimize the overall
global cost function CF(kb.sub.1,ku.sub.2,kp.sub.1, kM.sub.2, . . .
). A number of optimization algorithms are used in the software to
perform the optimization (e.g. Leven Berg Marquardt, Simulated
Annealing, . . . etc.).
[0304] To minimize the global cost function, the rate constants are
perturbed away from the starting values and the simulation is
repeated and the cost recalculated. If the cost is lower, the new
set of rate constants that gave the lower `cost` and a better fit
to the data are taken. Perturbing or changing of the rate constants
may be carried out almost randomly or more scientifically,
depending on the optimization routine. The process of changing the
rate constants, simulating the network, and evaluating the change
in the `cost` is repeated until the simulation nearly matches the
data. Parameter values are sought that give a simulation output
that matches the data. After optimization, the kinetic parameters
are constrained such that the simulation output matches the
experimental time points as shown in FIGS. 33(a)-(d) where the
solid line represents the simulation output.
[0305] In this way parameter values were constrained to fit the
measured data and the model was then used to predict cellular
behavior.
[0306] Predictions from the Model of the Physiological State of the
Cell and Finding Combination Therapies to Reverse the Cancer
State
[0307] The model predicts physiological outcomes such as
proliferation, G1-S, G2-M, S phase arrest, and apoptosis as
indicated by molecular markers within the simulation. For example,
CyclinE-CDK2 levels are an indicator of the G1-S state, CylinA-CDK2
levels are an indicator for the S phase state, CylinB-CDK1 levels
are an indicator for the G2-M state, and caspase 3 and cleaved PARP
(a protein that gets cleaved by executioner caspases such as
caspase 3) are indicators for apoptosis. The model is used to
simulate the "cancer" state indicated by high levels of
proliferative signals such as Erk and high levels of pro-apoptotic
proteins such as Bcl2. Targets can be perturbed to see if a
particular physiological state can be induced. After perturbing the
target one can predict whether the cells will go through G1-S
arrest, G2-M arrest, S phase arrest and apoptosis.
[0308] One can also use the model to predict the cell's sensitivity
to being in a particular state, e.g. sensitivity to apoptosis. By
way of explanation, normal cells express a certain number of
anti-apoptotic proteins. These proteins are analogous to actively
applying the brakes in a car at the top of a hill to prevent it
from rolling down, or, in the case of the cell, going into
apoptosis. If the brakes are released the car will not move forward
unless another force is applied, but without the brakes it is much
easier to send the car down the hill. Similarly, in a sensitized
state, the cell is more likely to go into apoptosis when another
perturbation that is pro-apoptotic is applied than when it is not
in a sensitized apoptotic state.
[0309] FIG. 34 shows the results of perturbing 41 individual
targets in the model where the final outcome is the cellular
physiological state of the cell. 41 targets were perturbed in the
simulation of the cancer cell. A perturbation was applied singly
either on the protein or RNA level such that the final outcome was
up or down regulation of the target on the protein level. This
perturbation can be accomplished systematically or automatically
via a computer algorithm that systematically perturbs each
component and then checks to see what state the cell is in.
[0310] Most of the perturbations shown in FIG. 34 lead to a more
sensitized apoptotic state. This is characteristic of a lot of
cancer therapeutics which have a single target as a component of
the diseased cell. Other putative therapeutics can be found by
determining the effect of their action on another node of
intervention. Within the simulation one can knock out a combination
of targets and thereby identify which combination, when knocked
out, is more likely to promote apoptosis.
[0311] FIG. 35 lists combinations of targets that were identified
by the simulation which when knocked out caused apoptosis in a
colon cancer cell. Surprisingly, it was found that many targets
when inhibited singly lead to sensitization towards apoptosis or
weak induction of apoptosis, but not apoptosis or a strong
induction of apoptosis. The combinations of targets synergistically
give rise to apoptosis in a cancer cell. For example, when one
inhibits Bcl2 or Bcl-xl in combination with CDK1, one can predict
apoptosis in the cancer cell. In contrast knocking out these
targets singly results in little or no caspase 3 activation.
Without being bound by theory, the mechanism for this may be that
inhibiting CDK1 in cells that are quickly dividing leads to high
levels of free CyclinB. This can sequester 1433-sigma away from the
pro-apoptotic proteins Bax and Bad, freeing them up to target the
mitochondria. This effect can further be enhanced by inhibiting
Bcl2 or Bcl-xL in combination. High levels of Bax and Bad can then
promote apoptosis via breakdown of the mitochondrial integrity.
[0312] FIG. 36 shows the mechanism of action of this perturbation
as described in the colon cell simulation. It is noted that FIG. 36
depicts a portion of FIGS. 33(f) and 33(g). In cells that are
dividing or quickly dividing (e.g. cancer cells), CDK1 can be
inhibited leading to induction of free CyclinB. CyclinB can then
bind 1433-sigma and sequester it away from Bad, Bax, and other
pro-apoptotic Bcl family members.
[0313] Identifying Key Nodes of Intervention from the
Simulation
[0314] The simulation is used to understand the conditions under
which oncogenic Ras leads to sustained levels of phosphoyrlated
Erk. Based on this analysis one can locate key places in the
network that drive sustained Erk levels and this knowledge can then
be used to identify new targets for therapeutic intervention.
[0315] Mutations in Ras lead to reduced hydrolysis rate such that
GAP is unable to efficiently convert GTP bound Ras to GDP bound
RAS. This mutation was explicitly put in the simulation by reducing
the parameter value controlling Ras hydrolysis rates. A cancerous
state is characterized by the cell's ability to create sustained
Erk levels with small levels of growth factors. Thus the system was
simulated under conditions where low levels of growth factors were
added. It was found that the only way one could attain sustained
levels of phosphorylated Erk, was where there was autocrine
signaling in the system, i.e. a feedback loop wherein Erk can lead
to the transcription of growth factors that bind to the EGF
receptor and further stimulate the network in an autocrine manner.
Thus another node or region where therapeutic intervention can be
helpful was discovered.
[0316] One can also perturb Ras or Erk directly and also perturb
the nodes that are responsible for the autocrine signaling and
thereby reverse the cancer state. FIGS. 37 through 38 show the
simulation output of this study. The upper graph depicts Erk
stimulation in the normal cell where it is stimulated transiently.
The lower graph above shows sustained Erk levels arising as a
result of having both an oncogenic Ras and autocrine signaling.
[0317] Testing Drugs/Compounds within the Model
[0318] A particular cancer state in which several nodes are mutated
has been simulated. The mutations in the model were inputted by:
(1) deleting beta-catenin Axin interaction so that high levels of
beta-catenin accumulate in the nucleus and transcription of cell
cycle target genes ensues; (2) mutating the Ras-Map kinase pathway
by reducing the kinetic parameter representing the GTP hydrolysis
rate of Ras and thus promoting high levels of erk and survival
signals; and (3) increasing the expression of bcl2 leading to
higher levels of the anti-apoptotic protein in the cell as is found
in many colorectal cancers.
[0319] An anti-sense bcl2 inhibitor G3139 (e.g. G3139, Benimetskaya
et al. 2001) currently in clinical trials was tested within the
model to determine whether it had any efficacy against the cancer
state. G3139 has been shown to reduce the total levels of Bcl2 over
24 hours. Using the simulation it was found that the cells become
sensitized to apoptosis, but most do not undergo apoptosis. The
sensitivity to apoptosis depends on the level and activity of
pro-apoptotic proteins such as Bax, Bik, and Bok. Cancers that have
mutations in these proteins are less sensitive to G3139
therapy.
[0320] It was also found that massive apoptosis can be induced if
the microenvironment of the cancer cell contains cytokines such as
TNF-alpha and FASL. These cytokines will further induce autocrine
and paracrine signaling that will promote apoptosis in the cancer
cell and surrounding cells. The cells become even more highly
sensitized when non-specific effects of G3139 were simulated, e.g.
its ability to inhibit XIAP and Bcl-xL mRNA translation
(Benimetskaya et al. 2001). To induce strong apoptosis requires a
secondary agent that leads to up-regulation of pro-apoptotic
proteins or cytokines. G3139 in and of itself has low toxicity.
Without being bound by theory, one can predict that the toxicity
induced with the secondary agent depends on the specificity of the
secondary agent to cancer cells. The more specific, the less toxic
the combination therapy will be. FIGS. 39-41 shows the Inhibition
of Bcl2 using G3139 antisense therapy.
[0321] This simulation illustrates how one skilled in the art can
use the model to simulate the treatment of a patient with a
specific stage of cancer by using one or more specific agents that
target key proteins controlling cell cycle progression and
apoptosis. The methods of invention demonstrate how one can use the
model to determine the efficacy of an agent against cancer with a
certain mutational profile. The methods of the invention include
optimization of the model with patient data and the use of the
model to analyze the best treatment strategy. The methods of the
invention determine how one can assess toxicity effects from using
a single or combination agent therapy on the normal cell.
[0322] Iterative Refinement of the Model Through Experiments The
colon cancer model as optimized above, predicted that inhibiting
NFkappaB and stimulating with TNF, in combination, would
synergistically promote increased levels of apoptosis. This
prediction was based on the fact that TNF concurrently activates
the apoptosis pathway via capase 8, 3, and 9 and the survival
pathway via IKK and NFkappaB. The latter pathway leads to
upregulation of survival signals which thwart or inhibit the
apoptotic signaling.
[0323] This experiment was performed on HCT116 to see if the
prediction was valid. FIG. 42 shows the cleavage of PARP as a
result of inhibiting Ikappab-alpha in combination with adding TNF
at various levels. FIG. 42 shows the relative levels of cleaved
PARP after 24 hours of stimulation with various doses of TNF. The
cells were stimulated with TNF alone, with 20 and 50 uM of the
Ikappab-alpha inhibitor, and with DMSO, the agent that solublizes
the inhibitor as a control. The inhibition of Ikkappab-alpha
effectively blocks NFkappaB nuclear translocation preventing
NFkappaB from activating the transcription of anti-apoptotic genes.
The cleavage of PARP signifies caspase 3 activation and to some
extent the degree of apoptosis in the cells. Surprisingly, it was
found that the combination did not synergize the promotion of
apoptosis. Upon further study it was found that in HCT116 cells,
NFkappaB accumulation in the nucleus in response to TNF was
transient and down stream anti-apoptotic targets did not get
activated. Thus in HCT116 cells the prediction is that the
combination therapy will not have a strong effect. This method can
be used to predict which cells would be responsive to TNF and
NFKappaB inhibition. Only those cells in which NFkappaB signaling
is strong as a result of TNF activation will respond strongly to
the combination. In this way, the model was refined and in its
refined state could be used to determine which cell types or
cancers could be treated by inhibiting NFkappaB transcription in
combination with adding TNF so as to synergize the promotion of
apoptosis. Exemplary method for determining validity and robustness
of the prediction
[0324] Since experimental data has associated uncertainties as
shown in FIG. 33(a) through 33(d), any parameters inferred by
fitting experimental data will also have uncertainties. There are
two types of uncertainty that may arise in this manner:
[0325] (1) Uncertainty in determining the parameter values around
one local minimum of the cost function, and
[0326] (2) Uncertainty in comparing different local minima of the
cost function.
[0327] Standard formulae for the propagation of uncertainties from
an experimental measurement to a set of fitting parameters apply to
case (1). These come from the standard error propagation
formula
(Delta K){circumflex over (
)}2=Sum.sub.--i(dK/dE.sub.--i){circumflex over ( )}2(Delta
E.sub.--i){circumflex over ( )}2
[0328] where K is an inferred parameter, E_i(Delta E_i) are the
experimental measurements (uncertainties), and Delta K is the
uncertainty in K.
[0329] For case (2), this formula does not apply. Case (2) arises
in the event that there are different sets of dynamical system
parameters that fit the data equally well, given the uncertainties
in the experimental data. The dynamical system can be used to
predict which experimental measurement would be most effective in
determining which set of dynamical system parameters is a more
accurate description of biology. This proceeds as follows:
[0330] (a) Simulate the different dynamical system parameters for
some randomly chosen initial conditions;
[0331] (b) Calculate for each unobserved molecule M, the root mean
square deviation over these randomly chosen initial conditions, for
the different dynamical system parameters defined as sigma_M;
[0332] (c) Check that the sensitivity of the parameters K in the
model which differ between the different dynamical system
parameters, with respect to measurements of levels of the molecules
E_M;
[0333] (d) Experimentally measure the level of the molecule with
the highest sum of total sensitivity and root mean square
deviation;
[0334] (e) Repeat the experimental fitting for all parameters to
check that some of the different parameters for local minima are no
longer local minima.
[0335] This iterative procedure eliminates systematically the
different local minima that arise in fitting experimental data to
large dynamical systems.
[0336] This methodology is useful in discerning between the
different predictions that a model might output as a result of
being underconstrained. For example, in the apoptosis pathway, the
onset of caspase activation shown in FIG. 32(g) can be due to a
heavy weighting of parameter values responsible for the positive
feedback loop between Caspase 8, Caspase 9, and Caspase 3 shown in
FIG. 32(g) or can be due to a heavy weighting of parameter values
responsible for autocrine signaling whereby the component JNK
further activates cytokines TNF and FASL. In other words, the above
methodology would output two fits to the data one with chemical
concentrations and parameter sensitvities that favor the feedback
mechanism between the caspases and the other that favored high
levels of TNF and FASL resulting from autocrine signaling. In the
former case, one would predict that perturbing the autocrine loop
would not have any consequence on inducing apoptosis in the cell
and in the latter case one would predict that it would. In this
case, the only way to distinguish between the two hypothesis
predicted by the model is to carry out an experiment that perturbed
the chemicals and or parameters deemed important by the above
analysis. In this way the methodology has been used to generate
more than one possible hypothesis. Our goal is to devise a method
to distinguish between the hypothesis or to determine where and how
the modul is robust. Using this methodology, an experiment can then
be designed to perturb the chemicals and or parameters showing the
greatest deviation and sensitivity between the hypothesis. The
experiment that is devised in this case
[0337] An Exemplary Modeling Tool: the Diagrammatic Cell Language
("DCL")
[0338] The Diagrammatic Cell Language is a language for modeling
and simulating biological systems. At its core is the notion that
the behavior of biological systems may be best understood as an
abstract computation. In this view point, the units of biological
heredity are packets of information, and the cell's biochemical
circuitry as a layer of computation evolved with the goal of
replicating the data stored in the hereditary material.
[0339] The Diagrammatic Cell Language is a precise representation
designed to be translated into a computer model of the reactions it
represents. This translation is referred to as parsing. The parsing
can occur via human modlers or a computer algorithm. The is the
beauty of the language: biologists can use it to map out describe
the biological interactions, modlers can then parse the
diagrammatic model built by the biologist and create a dynamical
simulation using said simulation environments (mention all other
simulation environments). The translation of the Diagrammatic Cell
Language into computer code using a computer algorithms has been
described in the DCL Provisional Patent.
[0340] This language is the best method for concisely representing
the massive amounts of cellular interactions that can occur in the
cell. This has been used to map the biological interactions of a
200 component network of colon a cell modularly represented in
FIGS. 31 and 32. The mathematical equations representing that
diagram have been parsed by a human modeler and are represented in
Exhibit A. What makes the language so efficient, are higher order
structures that can compactly represent all of the interactions a
cellular component can under go (e.g. a protein binding to multiple
components and getting modified by other entities). The constructs,
unique to the language: linkboxes and likeboxes are shown in FIG.
43(a), allow the biologist to compactly create diagrams of the cell
without simplifying the functional representation of a compound or
reaction. The language is modular: compounds and reactions may be
represented at levels of complexity at the discretion of the
biologist. It is object-oriented: a bound form of compounds, for
example, inherits the states of its constituents.
[0341] The in addition to such new constructs as linkboxes and
likeboxes, basic "noun" (single chemical entity such as the atom in
FIG. 43(a), "verb" (indicate transformations of a set of nouns to a
set of nouns), and "reaction/component shortcuts" are utilized. The
language constructs are summarized as follows (note: a formal
definition of the language is provided in DCL Provisional Patent
(see definitions for details) which includes a description of the
Diagrammatic or pictorial representation described here, the
parsing of the language into computer code, and an abstract
representation that uniquely defines the language):
[0342] Basic Grammatical Constructs
[0343] Atom--An atom is a noun. It is one particular state of a
molecule (or a molecule that is modeled as having one state). FIG.
43(a) depicts an example atom, A. (See FIG. 43(a)-43(b) for
examples of the graphic symbols in DCL.
[0344] Reaction--A reaction is a verb. It is a symbol that
represents a transformation of a set of nouns to a set of nouns.
Its symbol is a squiggle.
[0345] Dimerization--A dimerization is a shorthand symbol that
represents two compounds reacting with each other to form a new
bound form. Its symbol is a black dot:
[0346] Compartments--A compartment is a location with the cell. For
instance, as shown in FIG. 43(a), the nucleus (A is in the
nucleus):
[0347] New Grammatical Constructs
[0348] Linkbox--A linkbox is a noun that represents a collection of
states. It could be used to automatically represent, for example,
the many states of a protein that has several phosphorylation
sites, a protein that can undergo modifications, or a promoter. The
example linkbox is shown with with two states:
[0349] Likebox--A likebox is a noun that represents a group of
nouns or reactions that all have a similar function. An example of
a likebox which contains molecule A and molecule B (which act
alike) appears in FIG. 43(a).
[0350] Resolution Notation--Resolution notation is used to identify
particular states or subsets of states of a linkbox or likebox. The
basic form is a text string like this, which indicates that state
(1) is bound, and states (2-4) are unbound:
[0351] (1,0,0,0)
[0352] Equivalence line--This is a line that connects two nouns
that are equivalent. This indicates protein A is equivalent to
protein B.
[0353] Complex--A complex represents a collection of nouns that
bind together in a prescribed way. This is a complex of proteins A,
B and C (A binds to B, which binds to C, which in turn binds back
to the original A). See top of FIG. 43(b).
[0354] Modifier--Modifiers are used to express molecules that are
both reactants and products of a reaction, or inhibitions. They
could be used to represent catalysts, for example (C enables the
reaction of A to B).
[0355] Process--A process is a verb. It is a module which describes
either unknown or coarsely modeled behavior. The process is a
double-squiggle. Here is a process that could represent, coarsely,
the production of B from A.
[0356] Unique--The word unique describes a noun. It represents a
molecule that only appears once. An example is a Promoter. A unique
noun has a double-lined border.
[0357] Ubique--The word ubique describes a noun. It represents a
molecule that is so common it can be considered to be of constant
concentration and continuous availability. An example is Calcium. A
ubique noun has either a dashed border or no border.
[0358] Common--The word common describes a noun. It represents a
molecule that is neither unique or ubique, several copies of it may
exist in the cell. A common noun has a single-line border.
[0359] Below is a comparison between two approaches to describing a
portion of the Ras Activation pathway. The first approach (a) is a
more traditional approach to drawing a network diagram. It consists
solely of low-level language constructs. Notice how the (a)
diagram, alone is imprecise: it requires additional text to explain
the function of the molecules and thus can not efficiently be
converted into a set of mathematical equations for simulation.
[0360] (a) FIG. 44(a) depicts a portion of the Ras Activation
pathway expressed in a traditional way. The traditional depiction,
however, is unclear. The following text is required for
clarification:
[0361] Farnesyl protein transferase (FPTase) catalyzes the addition
of a farnesyl group onto C-N-Ras, C-K(A)-Ras, and C-K(B)-Ras.
Faresyl transferase inhibitor (FTI) blocks this reaction. Some
classes of FTIs work by irreversibly binding to FPTase.
[0362] Geranylgeranyl transferase I (GGTaseI) catalyzes the
addition of a geranylgeranyl moiety onto the same group of Ras
molecules. After lipid modification, each Ras molecules
translocates from the cytosol to the membrane.
[0363] (b) On the other hand, FIG. 44(b) depicts the same pathway
expressed with the Diagrammatic Cell Language. Provided that one
learns how to read the language, all of the above information in
the traditional depiction is contained solely within the DCL visual
diagram. Note how no additional text is necessary to explain the
function of the molecules, the diagram is much more concise, it is
precise, and it is much easier to read. The above diagram can then
be parsed into a distinct set of chemical states and a distinct set
of reactions. For each reaction the modler or computer algorithm
will have to attach a kinetic form and kinetic values to the
parameters. The above pictures parses into the following set of
states or chemical entities (the underscore here indicates the
modified or bound form of the chemical) and reaction steps which
the modeler can then attatch a specific kinetic form to:
5 Chemical Species Reaction Steps FPTase GGTase1 Gernyslates
C-N-Ras FTI FPTase Farnyslates C-N-Ras c-N-Ras_G C-N-Ras_F
translocates to the membrane c-N-Ras_F C-N-Ras_G translocates to
the membrane c-N-Ras_G_F C-N Ras_G_F translocates to the membrane
c-K(A)-Ras_G C-N-Ras translocates to the membrane c-K(A)-Ras_F .
c-K(A)-Ras_G_F . c-K(B) Ras_G . c-K(B)-Ras_F . c-K(B)-Ras_G_F
iterate these steps with c-K(A)-Ras and c-K(B)-Ras in place of
C-N-Ras. FPTase_FTI GGTase1 c-N-Ras_G_membrane c-N-Ras_F_membrane
c-N-Ras_membrane c-K(A)-Ras_G_membrane c-K(A)-Ras_F_mebrane
c-K(A)-Ras_mebrane c-K(B) Ras_G_membrane c-K(B)-Ras_F_membrane
c-K(B)-Ras_membrane
[0364] Exemplary Software Implementation for the Simulation and
Optimization of Biochemical Reaction Networks
[0365] It is exceedingly difficult to implement the methods of the
present invention without utilizing the data processing power of a
computer (in the most general sense of the term as a nonhuman
automated computational device) to solve the numerous equations
and/or formulae used in modeling and optimizing the models for
biochemical networks. Thus, in a preferred embodiment of the
invention, computer code is utilized to implement the present
invention. Such code can take many forms, and be implemented in a
variety of languages utilized for such purposes. The following
describes various exemplary embodiments of such code which were
written to implement the various functionalities of the present
invention.
[0366] A First Embodiment
[0367] Goals:
[0368] 1) To study the evolution of biochemical pathways in a
single cell, as observed in the simulation-generated time evolution
of various chemical species (proteins, enzymes, mRNA, etc.), given
a certain set of initial conditions, i.e., the initial
concentrations of each of the different species.
[0369] 2) To determine, within some bounds of error, the unknown
values of various pathway parameters, namely, the rate constants,
which are used to define the rate of reactions in the pathway.
[0370] Elements of the Code:
[0371] The code was written in C++, with network classes
representing the reaction networks, a director class to handle
multiple copies of the network, a minimizer class to determine
optimizable parameters and integrator classes to simulate the
behavior of the network over time. Chemical, reaction and rate
constant classes were defined to be used in the network class.
[0372] To optimize any given biochemical reaction network the user
had to define a network class specifying three lists of objects.
The first was a list of chemical objects corresponding to the
chemicals participating in the network. The second was a list of
rate constant objects that were used to define the rate of the
reactions. The third list consisted of reaction objects, each of
which used subsets of the chemical and rate constant list defined
above. The initial values of the chemicals and rate constants were
hard coded into this class.
[0373] To simulate multiple experiments on the same network,
multiple copies of the network would be defined, with different
initial conditions, and a director class would handle the list of
these networks. In the case where only one network was optimized,
the list in the director contained that single network. The
integrator to use was also specified directly in the director
class. The time series of chemicals that were to be plotted were
also specified in this class.
[0374] Method of Use:
[0375] To optimize the values of all rate constants specified in a
network the user had to go through the following steps, modifying
the code and recompiling for each run.
[0376] The user first had to create the network classes with the
appropriate chemicals, rate constants and reactions. Then he/she
would create the director class using these network classes. The
integrator to use for each network was specified here, and the
various variables controlling the behavior of the integrator were
also set in this class. The files containing the experimental time
series data against which optimization was done, was set in this
class. Then in the main body of the code, the director class, and a
minimizer class were instantiated, and the director was passed to
the minimizer for optimization of the rate constants.
[0377] The basic steps for the parameter optimization algorithm, as
found in most cases of optimization, are as follows:
[0378] 1) Time series for the chemical concentrations of each
chemical in each network class are generated. To do this, a set of
differential equations is generated by looping over the chemicals,
and the reactions that they participate in. Each differential
equation corresponds to the rate of change of the concentration of
a chemical in the network. The time series are generated by
integrating this set of diff. equations, starting with the initial
concentrations and the rate constants.
[0379] 2) Now the available experimental time series data for
chemicals in the pathway are compared to the corresponding
generated time series. The sum of squares of the differences
between the experimental concentration data and
simulation-generated concentration data at time points provided by
the experiment gives a value for the "objective function" for the
network.
[0380] 3) The "global objective function" value is computed by the
director by summing over the objective function value from each
network. The optimizable parameters, i.e., the rate constant
values, and the calculated global objective function value are then
passed to the minimizer. The minimizer then returns a new set of
values for the rate constants.
[0381] 4) Using the new rate constant values the networks are
reinitialized, and the code repeats the above steps. While
iterating through these steps, the minimizer tries to find
parameter values, so as to minimize the objective function. The
parameter values thus determined are then considered to best
describe the biochemical reaction network, giving sets of time
series that are closest to the experimental data for the user
provided initial concentrations.
[0382] Among the algorithms for integration of a set of ordinary
differential equations, this version included the well known
Runge-Kutta integrator and variations thereof, that allowed for
different levels of accuracy as well as dynamic adjustment of the
integration step-size for balancing speed and accuracy. The other
type of integration algorithm was an implementation of the
stochastic integrator by Gillespie that determined the cumulative
probability of occurrence of each reaction in the system and chose
one based on a uniform random number that is compared to the
cumulative probability. The optimization algorithms used were the
deterministic Levenberg-Marquardt method and the stochastic
Simulated Annealing method.
[0383] Problems Encountered with the First Embodiment of the
Computer Implementation:
[0384] 1) To simulate/optimize different biochemical networks, the
user had to setup new classes for each network, and the
corresponding director and then recompile the whole code. Even to
modify a single value for the initial chemical concentrations, rate
constants, integrator variables or minimizer variables, the code
had to be recompiled. This implied that a compiler had to be
available to the user and that the user had some knowledge of C++,
and compilation procedures.
[0385] 2) The way the code was setup either all or none of the rate
constants were optimized. The same was true for the initial
chemical concentrations, although the code had not been tested with
optimizable initial concentrations. In many cases, we would like to
optimize only a small subset of rate constants and initial chemical
concentrations.
[0386] 3) It was not possible to specify concentrations as
mathematical functions of other chemicals. This was often needed
since many experimental data consisted of time series of sums of
chemical concentrations. The rate constants were not variable at
all.
[0387] 4) Bounds could not be specified on the optimizable
parameters.
[0388] 5) When optimizing over multiple networks, the networks
class had to be exactly the same except for the initial
concentrations of the chemicals.
[0389] 6) Even though the code had very little graphical output, it
was tied to the Windows platform.
[0390] 7) Besides the above-mentioned problems, there were obvious
problems in regards to it's efficiency, and extensibility. It was
very hard to introduce prewritten libraries of minimizers and
integrators into the code. Also the code was not
parallelizable.
[0391] The Preferred Embodiment:
[0392] Goals:
[0393] In addition to the goals stated above, code was created to
eliminate the problems first encountered, as described above.
[0394] Elements of the Code:
[0395] The user defines a biochemical network by creating a text
file for the describing the network. Hence, the network class is
defined generally, and is filled in during runtime, based on the
description file. In addition, the chemicals and rate constants can
be made optimizable selectively through this file.
[0396] The director for multiple networks is specified in a
separate file. The networks, over which parameter identification is
to be done, are included in the director by including their file
names in the director file. The integrators and identifiers are
also defined through this file. Now a sequence of identifiers can
be used by specifying multiple identifiers in the input file.
[0397] By introducing input via the text files (see example), the
requirement for the user to have access to a compiler is
eliminated. Thus there is no requirement that a user have any
knowledge of C++. This also allows for a easier platform
independent architecture in regards to user input.
[0398] In the preferred embodiment, chemical concentrations and
rate constants are treated equally, as variables defining the state
of the network. If the user chooses, they can now be defined as
mathematical functions of other variables (both chemical
concentrations and rate constants) and numeric constants. They can
also be sent into the code with a predefined time series. To take
care of all these changes the mechanism for handling time series
was completely changed. A hierarchy of various time series classes
to handle the various types of definitions for chemicals and rate
constants was created. The form of the time series is specified in
the input file.
[0399] Summarizing the above improvements, where network variables
imply both chemicals and rate constants--
[0400] Network variables can be explicitly written as functions of
time.
[0401] Network variables can be expressed as mathematical functions
of other Network variables.
[0402] Rate constants can be unknowns, i.e., they can be treated as
state variables for which there is a differential equation to
integrate, just like the chemical state variables.
[0403] Network variables in the network can include time delays, so
they can depend on states of network variables at different
times.
[0404] Network variables can be expressed as switches, allowing
reactions to be turned on/off at specified times, or depending on
the state of some other network variable.
[0405] Time series expressing these network variables can be
expressed in terms of
[0406] a cubic spline
[0407] a polynomial interpolation
[0408] a mathematical formula--sum, product, difference, quotient,
power, elementary function (sin, cos, tan, arcsin, arccos, arctan,
log, exp), switch or gaussian.
[0409] Chemical concentrations can be either continuous or
discrete.
[0410] The code enables parallelization of the code using the MPI
(Message Passing Interface) library. The simulated annealing
parameter identifier was completely rewritten, allowing for the use
of a parallel architecture.
[0411] The preferred code now supports the integration of "stiff"
systems of differential equations, resulting from reactions
occurring on very different time scales. A stochastic integrator is
included based on the "Next-Reaction" method by Gibson, which is
more efficient than the previously used Gillespie algorithm.
[0412] The preferred code can compute the sensitivity of the cost
function to changes in the parameters. The cost function
sensitivities are computed by solving the sensitivity equations,
which give the sensitivity of each chemical concentration with
respect to the parameters. The chemical sensitivities are used to
compute a raw value for the cost function sensitivity. Since these
raw values depend on the scale of the associated parameters, they
are normalized so that the sum of the squares of the normalized
values is equal to one. With the normalized values, one can
determine immediately which parameters have the greatest effect on
the cost function; i.e., on the goodness of fit of the model.
[0413] The parameter identification has as also been improved. The
new simulated annealing algorithm has been parallelized. The
preferred code introduces a separate temperature schedule to
gradually limit the range in parameter space, over which the next
step can move. The deterministic Levenberg-Marquardt can use the
parameter sensitivities to compute the gradient. This is both more
efficient and more accurate than a finite difference approximation.
Both parameter identifiers now allow for imposing a lower and upper
bound on each optimizable parameter, thus narrowing the parameter
search space.
[0414] Other features of the preferred code include a new parameter
identification engine that has
[0415] Optimizable parameters can be selected based on
sensitivities.
[0416] An adaptive minimizer which iteratively chooses a subset of
the parameters to make optimizable.
[0417] An analysis tool to aid the modeler in determining which
parameters have the greatest effect on the concentration of
specified chemicals.
[0418] When solving a system of stiff ODEs, it is necessary to
solve a system of nonlinear equations at each time step. The
solution of this nonlinear system uses Newton's method which is an
iterative method that requires solving a linear system using the
Jacobian matrix at each step. For most problems the Jacobian matrix
is sparse, and the code has been written to take advantage of this
sparsity. The linear system can be solved using either a
preconditioned Krylov method or a banded solver. The new code has
the ability to reorder the state variables in such a way that the
bandwidth of the Jacobian matrix is reduced, making the solution
much more efficient. For some problems, the speed-up in the new
code is as much as two orders of magnitude.
[0419] The basic set of steps to run an optimization are depicted
in the flowchart of FIG. 45.
[0420] Alternative Embodiment of the Preferred Code
[0421] Additional features of an alternative embodiment of the
preferred code include:
[0422] Object-Oriented Database: An object-oriented database (OODB)
to house data for various biochemical reaction networks. Using a
separate code the user will be able to use the database to setup a
simulation or parameter identification run. The input portion of
the code reads the database directly from the database and then
save the outputs back to it. The OO nature of the database
simplifies the task of communicating with the database since the
same classes for the code can be used, as those that are used to
define the database.
[0423] Improved parallel algorithms for optimization: In this
embodiment, the parallel simulated annealing code is improved by
implementing various parallel architectures in the code. The
Levenberg-Marquardt identifier is also be improved by a
straightforward parallelization of the algorithm.
[0424] Multi-threaded integration: The integration algorithms are
made multi-threaded. This leads to a better performance than using
only parallel code.
[0425] Hybrid method: Standard intergrations methods are either
purely continuous or purely stochastic. For many of the large
systems contemplated to be analyzed by the methods of the present
invention, a purely stochastic algorithm is too slow. On the other
hand a purely continuous algorithm inaccurately describes reactions
involving chemicals with few molecules in the system. Very often a
biochemical network contains some molecules which play a central
role, yet they have a very occurrences of them in the system. Hence
to properly yet efficiently simulate a system a hybrid
stochastic-continuous integrator is optimal.
[0426] Among other parameter identification algorithms, genetic
algorithms and the direct fit method are implemented.
[0427] Example Showing the Use of the Preferred Code:
[0428] The following example shows how users can input models into
the software and specify the integration and optimization
algorithms to use. The first file the code reads is the director
file. This file specifies the network(s), the integrator and the
method(s) to be used for parameter identification.
6 director = p53_director { network = p53_network { source =
"p53_network.txt"; }; integrator = cvode { relative_tolerance =
1e-5; final_time = 1200; }; }; parameter_identification = {
identifier = simulated_annealing { max_temp_scale_factor = 100
max_steps_per_temperature = 100; num_initial_trial_steps = 100;
temp_anneal_factor = 0.9; parameter_temp_anneal_factor = 0.95; };
identifier = levenberg_marquardt { max_iterations = 5; f_precision
= 1e-5; }; };
[0429] The director file contains a director block and a
parameter_identification block. The director block contains one or
more network blocks and an integrator block. In this example, the
director has one network which is read in from the file
"p53_network.txt". The integrator block specifies the integrator to
use (in this case CVode, which is a third-party package for
integrating stiff systems of ODEs) and parameters which control the
behavior of the integrator. The parameter_identification block is
optional; if it is omitted, the code will run a simulation and
exit. If included, it specifies one or more identifiers to run
along with options controlling their behavior. If multiple
identifiers are specified, the first will run with the initial
values of the parameters specified by the user, the second will
start with the optimized values found by the first identifier,
etc.
[0430] In keeping with the modular design, the network is usually
read in from a separate file as in this example. In addition,
different parts of the network (chemicals, parameters, reactions,
etc.) are typically contained in separate files as well. For this
example, the file "p53_network.txt" has the following lines:
7 includefile = "p53_inputchems.txt"; includefile =
"p53_chemsnpars.txt"; includefile = "p53_reactions.txt";
includefile = "p53_expdata.txt"; includefile =
"p53_display.txt";
[0431] These lines tell the code to read in the named files in the
order listed. The file "p53_inputchems.txt" has the following
lines:
8 chemicals = { damagedDNA = (0); E2F = (500); ChromosomeSignal =
(0); };
[0432] These lines specify the names of three chemicals and their
initial concentrations. Enclosing the initial concentrations in
parentheses tells the code that these chemicals are state
variables; i.e., their concentrations can be changed as the result
of some reaction in the network. The equal sign ("=") means that
the initial concentrations are fixed; they will not be changed in
the course of the optimization.
9 Next is the file "p53_chemsnpars.txt": chemicals = { One = 1.0;
DNAPK = (1000); Rad = (1000); ATR = (1000); ATM = (1000); Chk1 =
(1000); Chk2 = (1000); p53 = (50); CKI = (1000); CBP = (1000); ARF
= (1000); Mdm2 = (1000); XXXPromoter = (1.0); YYYPromoter = (1.0);
}; parameters = { kd_damaged DNA = 0.02; kp_DNAPK_a = 0.2;
km_DNAPK_a = 500; kt_DNAPK_i = 0.005; kp_Rad_a = 0.2; km_Rad_a =
500; kt_Rada_preG2 = 0.1; kp_Rada_postG2 = 10; km_Rada_postG2 = 10;
kt_Rad_i = 0.005; kp_ATR_a = 0.2; km_ATR_a = 500; kt_ATR_i = 0.005;
kp_Chk1_Phos = 0.2; km_Chk1_Phos = 500; kt_Chk1_Dephos = 0.005;
kp_ATM_a = 0.2; km_ATM_a = 500; kt_ATM_i = 0.005; kp_Chk2_Phos =
0.2; km_Chk2_Phos = 500; kt_Chk2_Dephos = 0.005; ks_p53 .about.
500; kd_p53 .about. 0.5; kt_p53_Nucl .about. 0.2; kt_p53_Cyto
.about. 0.2; kt_Mdm2_Nucl = 0.2; kt_Mdm2_Cyto = 0.2;
kt_Mdm2_p53_Nucl = 0.2; kt_Mdm2_p53_Cyto = 0.2; kp_p53_15Phos =
0.2; km_p53_15Phos = 500; kp_p5315Phos_15Phos = 0.2;
km_p5315Phos_15Phos = 500; kp_p53_20Phos1 = 0.2; km_p53_20Phos1 =
500; kp_p53_20Phos2 = 0.2; km_p53_20Phos2 = 500; kb_p53_CBP =
0.001; ku_p53_CBP = 0.5; kp_p53_37Phos = 0.2; km_p53_37Phos = 500;
ks_ARF = 500; kd_ARF = 0.5; kb_ARF_E2F = 0.001; ktd_E2F = 0.1;
kb_ARF_Mdm2 = 0.001; ku_ARF_Mdm2 = 0.5; kb_Mdm2_p53 = 0.01;
ku_Mdm2_p53 = 0.5; ktd_p53 = 1; kb_XXXPromoter_p53 = 0.001;
ku_XXXPromoter_p53 = 0.5; ks_XXXmRNA = 50; kt_XXXmRNA_Cyto = 0.1;
ks_XXX = 0.5; kd_XXXmRNA = 0.01; kd_XXX = 0.001;
kb_Mdm2Promoter_p53 = 0.001; ku_Mdm2Promoter_p53 = 0.5; ks_Mdm2mRNA
= 50; kt_Mdm2mRNA_Cyto = 0.1; ks_Mdm2 = 0.5; kd_Mdm2mRNA = 0.01;
kd_Mdm2 = 0.001; kb_YYYPromoter_p53 = 0.001; ku_YYYPromoter_p53 =
0.5; ks_YYYmRNA = 50; kt_YYYmRNA_Cyto = 0.1; ks_YYY = 0.5;
kd_YYYmRNA = 0.01; kd_YYY = 0.001; };
[0433] This file defines some more chemicals and some parameters.
The values of most of the parameters are specified by the equal
sign, so they will not be changed by the optimizer. The lines
10 ks_p53 .about. 500; kd_p53 .about. 0.5; kt_p53_Nucl .about. 0.2;
kt_p53_Cyto .about. 0.2;
[0434] indicate that these four parameters are optimizable; when
the optimizer runs, it will change the values of these four
parameters in an attempt to fit the experimental data.
[0435] Next the file "p53_reactions.txt" lists all the reactions
that define the network:
11 reactions = { UDR(damagedDNA.vertline.kd_damaged DNA);
MMR(damagedDNA,DNAPK,DNAPKa.vertline.kp_DNAPK_a,km_DNAPK_a);
TR(DNAPKa,DNAPK.vertline.kt_DNAPK_i);
MMR(damagedDNA,Rad,Rada.vertline.kp_Rad_a,km_Rad_a);
TR(Rada,RadpreG2.vertline.kt_Rada_preG2); MMR(ChromosomeSignal,Rad-
a,RadpostG2.vertline.kp_Rada_postG2,km_Rada_pos tG2);
TR(RadpreG2,Rad.vertline.kt_Rad_i); TR(RadpostG2,Rad.vertline.kt_R-
ad_i); MMR(RadpreG2,ATR,ATRa.vertline.kp_ATR_a,km_ATR_a);
TR(ATRa,ATR.vertline.kt_ATR_i); MMR(ATRa,Chk1,Chk1Phos.vertline.kp-
_Chk1_Phos,km_Chk1_Phos);
TR(Chk1Phos,Chk1.vertline.kt_Chk1_Dephos)- ;
MMR(RadpostG2,ATM,ATMa.vertline.kp_ATM_a,km_ATM_a);
TR(ATMa,ATM.vertline.kt_ATM_i); MMR(ATMa,Chk2,Chk2Phos.vertline.kp-
_Chk2_Phos,km_Chk2_Phos);
TR(Chk2Phos,Chk2.vertline.kt_Chk2_Dephos)- ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% p53 generation and
natural degradation %% %%%%%%%%%%%%%%%%%%%%%%%%-
%%%%%%%%%%%%%%%%%%%% GR(One,p53.vertline.ks_p53);
UDR(p53.vertline.kd_p53); UDR(p5337Phos.vertline.kd_p53);
UDR(p5320Phos.vertline.kd_p53); UDR(p5320Phos37Phos.vertline.kd_p5-
3); UDR(p5320Phos_CBP.vertline.kd_p53);
UDR(p5320Phos37Phos_CBP.vertline.kd_p53); UDR(p5315Phos.vertline.k-
d_p53); UDR(p5315Phos37Phos.vertline.kd_p53);
UDR(p5315Phos20Phos.vertline.kd_p53); UDR(p5315Phos20Phos37Phos.ve-
rtline.kd_p53); UDR(p5315Phos20Phos_CBP.vertline.kd_p53);
UDR(p5315Phos20Phos37Phos_CBP.vertline.kd_p53);
UDR(p5315PhosPhos.vertline.kd_p53); UDR(p5315PhosPhos37Phos.vertli-
ne.kd_p53); UDR(p5315PhosPhos20Phos.vertline.kd_p53);
UDR(p5315PhosPhos20Phos37Phos.vertline.kd_p53);
UDR(p5315PhosPhos20Phos_CBP.vertline.kd_p53);
UDR(p5315PhosPhos20Phos37Phos_CBP.vertline.kd_p53);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% nuclear shuttling of p53
and Mdm2 %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
TR(p53Cyto,p53.vertline.kt_p53_Nucl); TR(p53,p53Cyto.vertline.kt_p-
53_Cyto); TR(p5337PhosCyto,p5337Phos.vertline.kt_p53_Nucl);
TR(p5337Phos,p5337PhosCyto.vertline.kt_p53_Cyto);
TR(p5320PhosCyto,p5320Phos.vertline.kt_p53_Nucl);
TR(p5320Phos,p5320PhosCyto.vertline.kt_p53_Cyto);
TR(p5320Phos37PhosCyto,p5320Phos37Phos.vertline.kt_p53_Nucl);
TR(p5320Phos37Phos,p5320Phos37PhosCyto.vertline.kt_p53_Cyto);
TR(p5320Phos_CBPCyto,p5320Phos_CBP.vertline.kt_p53_Nucl);
TR(p5320Phos_CBP,p5320Phos_CBPCyto.vertline.kt_p53_Cyto);
TR(p5320Phos37Phos_CBPCyto,p5320Phos37Phos_CBP.vertline.kt_p53_Nucl);
TR(p5320Phos37Phos_CBP,p5320Phos37Phos_CBPCyto.vertline.kt_p53_Cyto);
TR(p5315PhosCyto,p5315Phos.vertline.kt_p53_Nucl);
TR(p5315Phos,p5315PhosCyto.vertline.kt_p53_Cyto);
TR(p5315Phos37PhosCyto,p5315Phos37Phos.vertline.kt_p53_Nucl);
TR(p5315Phos37Phos,p5315Phos37PhosCyto.vertline.kt_p53_Cyto);
TR(p5315Phos20PhosCyto,p5315Phos20Phos.vertline.kt_p53_Nucl);
TR(p5315Phos20Phos,p5315Phos20PhosCyto.vertline.kt_p53_Cyto);
TR(p5315Phos20Phos37PhosCyto,p5315Phos20Phos37Phos.vertline.kt_p53_Nucl)
; TR(p5315Phos20Phos37Phos,p5315Phos20Phos37PhosCyto.vertli-
ne.kt_p53_Cyto) ; TR(p5315Phos20Phos_CBPCyto,p5315Phos20Phos-
_CBP.vertline.kt_p53_Nucl);
TR(p5315Phos20Phos_CBP,p5315Phos20Phos_-
CBPCyto.vertline.kt_p53_Cyto);
TR(p5315Phos20Phos37Phos_CBPCyto,p53-
15Phos20Phos37Phos_CBP.vertline.kt_p5 3_Nucl);
TR(p5315Phos20Phos37Phos_CBP,p5315Phos20Phos37Phos_CBPCyto.vertline.kt_p5
3_Cyto); TR(p5315PhosPhosCyto,p5315PhosPhos.vertline.kt_p53-
_Nucl); TR(p5315PhosPhos,p5315PhosPhosCyto.vertline.kt_p53_Cyto);
TR(p5315PhosPhos37PhosCyto,p5315PhosPhos37Phos.vertline.kt_p53_Nucl)-
;
TR(p5315PhosPhos37Phos,p5315PhosPhos37PhosCyto.vertline.kt_p53_Cy-
to);
TR(p5315PhosPhos20PhosCyto,p5315PhosPhos20Phos.vertline.kt_p53-
_Nucl);
TR(p5315PhosPhos20Phos,p5315PhosPhos20PhosCyto.vertline.kt_-
p53_Cyto);
TR(p5315PhosPhos20Phos37PhosCyto,p5315PhosPhos20Phos37Ph-
os.vertline.kt_p5 3_Nucl); TR(p5315PhosPhos20Phos37Phos,p531-
5PhosPhos20Phos37PhosCyto.vertline.kt_p5 3_Cyto);
TR(p5315PhosPhos20Phos_CBPCyto,p5315PhosPhos20Phos_CBP.vertline.kt_p53_N
ucl); TR(p5315PhosPhos20Phos_CBP,p5315PhosPhos20Phos_CBPCyt-
o.vertline.kt_p53_C yto); TR(p5315PhosPhos20Phos37Phos_CBPCy-
to,p5315PhosPhos20Phos37Phos_C BP.vertline.kt_p53_Nucl);
TR(p5315PhosPhos20Phos37Phos_CBP,p5315PhosPhos20Phos37Phos_CBPCy
to.vertline.kt_p53_Cyto); TR(Mdm2Cyto,Mdm2.vertline.kt_Mdm2_Nucl);
TR(Mdm2,Mdm2Cyto.vertline.kt_Mdm2_Cyto);
TR(Mdm2_p53Cyto,Mdm2_p53.vertline.kt_Mdm2_p53_Nucl);
TR(Mdm2_p53,Mdm2_p53Cyto.vertline.kt_Mdm2_p53_Cyto);
TR(Mdm2_p5337PhosCyto,Mdm2_p5337Phos.vertline.kt_Mdm2_p53_Nucl);
TR(Mdm2_p5337Phos,Mdm2_p5337PhosCyto.vertline.kt_Mdm2_p53_Cyto);
%%%%%%%%%%%%%%%%%%%%%%%%%%% %% p53_15Phos by DNAPKa %%
%%%%%%%%%%%%%%%%%%%%%%%%%%% MMR(DNAPKa,p53,p5315Phos.vertline.kp_p-
53_15Phos,km_p53_15Phos);
MMR(DNAPKa,p5337Phos,p5315Phos37Phos.vert-
line.kp_p53_15Phos,km_p53_15Ph os); MMR(DNAPKa,p5320Phos,p53-
15Phos20Phos.vertline.kp_p53_15Phos,km_p53_15Ph os);
MMR(DNAPKa,p5320Phos37Phos,p5315Phos20Phos37Phos.vertline.kp_p53_15Phos,
km_p53_15Phos); MMR(DNAPKa,p5320Phos_CBP,p5315Phos20Phos_CB-
P.vertline.kp_p53_15Phos,km_p 53_15Phos);
MMR(DNAPKa,p5320Phos37Phos_CBP,p5315Phos20Phos37Phos_CBP.vertline.kp_p53.-
sub.-- _15Phos,km_p53_15Phos); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%- % %%
p5315Phos_15Phos by CKI %% %%%%%%%%%%%%%%%%%%%%%%%%%%- %%%%
MMR(CKI,p5315Phos,p5315PhosPhos.vertline.kp_p5315Phos_15Phos,k-
m_p5315Pho s_15Phos); MMR(CKI,p5315Phos37Phos,p5315PhosPhos3-
7Phos.vertline.kp_p5315Phos_15Phos, km_p5315Phos_15Phos);
MMR(CKI,p5315Phos20Phos,p5315PhosPhos20Phos.vertline.kp_p5315Phos_15Phos,
km_p5315Phos_15Phos); MMR(CKI,p5315Phos20Phos37Phos,p5315Ph-
osPhos20Phos37Phos.vertline.kp_p5315
Phos_15Phos,km_p5315Phos_15Pho- s);
MMR(CKI,p5315Phos20Phos_CBP,p5315PhosPhos20Phos_CBP.vertline.kp-
_p5315Phos _15Phos,km_p5315Phos_15Phos);
MMR(CKI,p5315Phos20Phos37Phos_CBP,p5315PhosPhos20Phos37Phos_CBP.vertline.
kp_p5315Phos_15Phos,km_p5315Phos_15Phos);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% p53_20Phos by Chk1 and Chk2
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
MMR(Chk1Phos,p53,p5320Phos.vertline.kp_p53_20Phos1,km_p53_20Phos1);
MMR(Chk1Phos,p5337Phos,p5320Phos37Phos.vertline.kp_p53_20Phos1,km_p53_2
0Phos1); MMR(Chk1Phos,p5315Phos,p5315Phos20Phos.vertline.kp-
_p53_20Phos1,km_p53_2 0Phos1); MMR(Chk1Phos,p5315Phos37Phos,-
p5315Phos20Phos37Phos.vertline.kp_p53_20Pho s1,km_p53_20Phos1);
MMR(Chk1Phos,p5315PhosPhos,p5315PhosPhos20Phos.vertline.kp_p53_20Phos1-
,k m_p53_20Phos1); MMR(Chk1Phos,p5315PhosPhos37Phos,p5315Pho-
sPhos20Phos37Phos.vertline.kp_p5 3_20Phos1,km_p53_20Phos1);
MMR(Chk2Phos,p53,p5320Phos.vertline.kp_p53_20Phos2,km_p53_20Phos2);
MMR(Chk2Phos,p5337Phos,p5320Phos37Phos.vertline.kp_p53_20Phos2,km_p53_2
0Phos2); MMR(Chk2Phos,p5315Phos,p5315Phos20Phos.vertline.kp-
_p53_20Phos2,km_p53_2 0Phos2); MMR(Chk2Phos,p5315Phos37Phos,-
p5315Phos20Phos37Phos.vertline.kp_p53_20Pho s2,km_p53_20Phos2);
MMR(Chk2Phos,p5315PhosPhos,p5315PhosPhos20Phos.vertline.kp_p53_20Phos2-
,k m_p53_20Phos2); MMR(Chk2Phos,p5315PhosPhos37Phos,p5315Pho-
sPhos20Phos37Phos.vertline.kp_p5 3_20Phos2,km_p53_20Phos2);
%%%%%%%%%%%%%%%%%%%%%%%% %% b+u_p5320Phos_CBP %%
%%%%%%%%%%%%%%%%%%%%%%%% HDR(p5320Phos37Phos,CBP,p5320Phos37Phos_C-
BP.vertline.kb_p53_CBP);
HDDR(p5320Phos37Phos_CBP,p5320Phos37Phos,C-
BP.vertline.ku_p53_CBP);
HDR(p5320Phos37Phos,CBP,p5320Phos37Phos_CB- P.vertline.kb_p53_CBP);
HDDR(p5320Phos37Phos_CBP,p5320Phos37Phos,CB-
P.vertline.ku_p53_CBP);
HDR(p5315Phos20Phos37Phos,CBP,p5315Phos20Ph-
os37Phos_CBP.vertline.kb_p53.sub.-- CBP);
HDDR(p5315Phos20Phos37Phos_CBP,p5315Phos20Phos37Phos,CBP.vertline.ku_p53.-
sub.-- CBP); HDR(p5315Phos20Phos37Phos,CBP,p5315Phos20Phos37-
Phos_CBP.vertline.kb_p53.sub.-- CBP);
HDDR(p5315Phos20Phos37Phos_CBP,p5315Phos20Phos37Phos,CBP.vertline.ku_p53.-
sub.-- CBP); HDR(p5315PhosPhosPhos20Phos37Phos,CBP,p5315Phos-
Phos20Phos37Phos.sub.-- CBP.vertline.kb_p53_CBP);
HDDR(p5315PhosPhos20Phos37Phos_CBP,p5315PhosPhos20Phos37Phos,CBP
.vertline.ku_p53_CBP); HDR(p5315PhosPhos20Phos37Phos,CBP,p5315Phos-
Phos20Phos37Phos_CBP.vertline. kb_p53_CBP);
HDDR(p5315PhosPhos20Phos37Phos_CBP,p5315PhosPhos20Phos37Phos,CBP
.vertline.ku_p53_CBP); %%%%%%%%%%%%%%%%%%%%%%%%%%% %% p53_37Phos by
DNAPKa %% %%%%%%%%%%%%%%%%%%%%%%%%%%%
MMR(DNAPKa,p53,p5337Phos.vertline.kp_p53_37Phos,km_p53_37Phos);
MMR(DNAPKa,p5320Phos,p5320Phos37Phos.vertline.kp_p53_37Phos,km_p53_37Ph
os); MMR(DNAPKa,p5320Phos_CBP,p5320Phos37Phos_CBP.vertline.kp-
_p53_37Phos,km_p 53_37Phos); MMR(DNAPKa,p5315Phos,p5315Phos3-
7Phos.vertline.kp_p5315Phos_37Phos,km_p53 15Phos_37Phos);
MMR(DNAPKa,p5315Phos20Phos,p5315Phos20Phos37Phos.vertline.kp_p5315Phos_3
7Phos,km_p5315Phos_37Phos); MMR(DNAPKa,p5315Phos20Phos_CBP,-
p5315Phos20Phos37Phos_CBP.vertline.kp_p531
5Phos_37Phos,km_p5315Pho- s_37Phos);
MMR(DNAPKa,p5315PhosPhos,p5315PhosPhos37Phos.vertline.kp-
_p5315PhosPhos_3 7Phos,km_p5315PhosPhos_37Phos);
MMR(DNAPKa,p5315PhosPhos20Phos,p5315PhosPhos20Phos37Phos.vertline.kp_p531
5PhosPhos_37Phos,km_p5315PhosPhos_37Phos);
MMR(DNAPKa,p5315PhosPhos20Phos_CBP,p5315PhosPhos20Phos37Phos_CBP
.vertline.kp_p5315PhosPhos_37Phos,km_p5315PhosPhos_37Phos);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%
p53 degradation through Mdm2, inhibition of that by Arf %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
GR(E2F,ARF.vertline.ks_ARF); UDR(ARF.vertline.kd_ARF);
TR(ARF_E2F,E2F.vertline.kd_ARF); TR(ARF_Mdm2,Mdm2.vertline.kd_ARF)-
; HDR(ARF,E2F,ARF_E2F.vertline.kb_ARF_E2F);
TR(ARF_E2F,ARF.vertline.ktd_E2F); HDR(ARF,Mdm2,ARF_Mdm2.vertline.k-
b_ARF_Mdm2); HDDR(ARF_Mdm2,ARF,Mdm2.vertline.ku_ARF_Mdm2);
HDR(Mdm2,p53,Mdm2_p53.vertline.kb_Mdm2_p53);
HDDR(Mdm2_p53,Mdm2,p53.vertline.ku_Mdm2_p53);
HDR(Mdm2Cyto,p53Cyto,Mdm2_p53Cyto.vertline.kb_Mdm2_p53);
HDDR(Mdm2_p53Cyto,Mdm2Cyto,p53Cyto.vertline.ku_Mdm2_p53);
HDDR(Mdm2_p53Cyto,Mdm2Cyto,dumpedp53.vertline.ktd_p53);
HDR(Mdm2,p5337Phos,Mdm2_p5337Phos.vertline.kb_Mdm2_p53);
HDDR(Mdm2_p5337Phos,Mdm2,p5337Phos.vertline.ku_Mdm2_p53);
HDR(Mdm2Cyto,p5337PhosCyto,Mdm2_p5337PhosCyto.vertline.kb_Mdm2_p53);
HDDR(Mdm2_p5337PhosCyto,Mdm2Cyto,p5337PhosCyto.vertline.ku_Mdm2_p53);
HDDR(Mdm2_p5337PhosCyto,Mdm2Cyto,dumpedp53.vertline.ktd_p53);
UDR(dumpedp53.vertline.kd_p53); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%-
%%%%%%%% %% various transcriptions caused by p53 %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% HDR(XXXPromoter,p5315Ph-
osPhos20Phos37Phos_CBP,XXXPromotera.vertline.kb_XX XPromoter_p53);
HDDR(XXXPromotera,XXXPromoter,p5315PhosPhos20Phos37Phos_CBP.vertli-
ne.ku_X XXPromoter_p53); GR(XXXPromotera,XXXmRNA.vertline.ks-
_XXXmRNA); TR(XXXmRNA,XXXmRNACyto.vertline.kt_XXXmRNA_Cyto);
GR(XXXmRNACyto,XXXCyto.vertline.ks_XXX); UDR(XXXmRNA.vertline.kd_X-
XXmRNA); UDR(XXXmRNACyto.vertline.kd_XXXmRNA);
UDR(XXXCyto.vertline.kd_XXX); HDR(Mdm2Promoter,p5315PhosPhos20Phos-
37Phos_CBP,Mdm2Promotera.vertline.kb.sub.-- Mdm2Promoter_p53);
HDDR(Mdm2Promotera,Mdm2Promoter,p5315PhosPhos20Phos37Phos_CBP.vertline.-
ku.sub.-- Mdm2Promoter_p53); GR(Mdm2Promotera,Mdm2mRNA.vertl-
ine.ks_Mdm2mRNA);
TR(Mdm2mRNA,Mdm2mRNACyto.vertline.kt_Mdm2mRNA_Cyt- o);
GR(Mdm2mRNACyto,Mdm2Cyto.vertline.ks_Mdm2);
UDR(Mdm2mRNA.vertline.kd_Mdm2mRNA); UDR(Mdm2mRNACyto.vertline.kd_M-
dm2mRNA); UDR(Mdm2.vertline.kd_Mdm2);
UDR(Mdm2Cyto.vertline.kd_Mdm2); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%-
%%%%%%%%%%% %% various transcriptions inhibited by p53 %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
HDR(YYYPromoter,p5315PhosPhos20Phos37Phos_CBP,YYYPromoteri.vertline.kb_YY
YPromoter_p53); HDDR(YYYPromoteri,YYYPromoter,p5315PhosPhos-
20Phos37Phos_CBP.vertline.ku_Y YYPromoter_p53);
GR(YYYPromoter,YYYmRNA.vertline.ks_YYYmRNA);
TR(YYYmRNA,YYYmRNACyto.vertline.kt_YYYmRNA_Cyto);
GR(YYYmRNACyto,YYYCyto.vertline.ks_YYY); UDR(YYYmRNA.vertline.kd_Y-
YYmRNA); UDR(YYYmRNACyto.vertline.kd_YYYmRNA);
UDR(YYYCyto.vertline.kd_YYY); };
[0436] Each reaction is specified by an abbreviation (e.g., HDR
means HeteroDimerization Reaction, MMR means Michaelis-Menton
Reaction, etc.) followed by a list of chemicals and rate constants
(parameters) that participate in the reaction. The chemicals are
listed first followed by a vertical bar (".vertline.") and the list
of rate constants. The chemicals and rate constants must be listed
in the correct order. For example the line
[0437] HDR(A,B,C.vertline.kb);
[0438] defines a reaction in which a molecule of A and a molecule
of B bind to form a molecule of C., Thus A and B can be listed in
either order, but C must be specified in the third position. The
lines in the above example that begin with a percent sign ("%") are
comments and are ignored by the code. Any chemical that is used in
a reaction without being defined in a chemicals block is created
with a default initial concentration of zero.
[0439] The file "p53_expdata.txt" contains the experimental data
used by the optimizers:
12 experimental_data = { data = p53 { data_type = value; values = (
(0, 50, 1), (20, 100, 1), (40, 70, 1), (80, 60, 1) ); }; };
[0440] In this case, there is experimental data for only one
chemical, p53. The values are specified in triples in which the
first number is the time of the observation, the second is the
concentration, and the third is the error.
[0441] Finally, the file "p53_display.txt" controls the output of
the code.
13 chemicals = { t_p5315Phos = SUM( p5315Phos,p5315Phos37Phos,
p5315Phos20Phos,p5315Phos20Phos37Phos,
p5315Phos20Phos_CBP,p5315Phos20Phos37Phos_CBP); t_p5315PhosPhos =
SUM( p5315PhosPhos,p5315PhosPhos37Phos,
p5315PhosPhos20Phos,p5315PhosPhos20Phos37Phos,
p5315PhosPhos20Phos_CBP,p5315PhosPhos20Phos37Phos_CBP);
t_p5320Phos_NoCBP = SUM( p5320Phos,p5320Phos37Phos,
p5315Phos20Phos,p5315Phos20Phos37Phos, p5315PhosPhos20Phos,p5315Ph-
osPhos20Phos37Phos); t_p5320Phos_CBP = SUM(
p5320Phos_CBP,p5320Phos37Phos_CBP, p5315Phos20Phos_CBP,p5315Phos20-
Phos37Phos_CBP,
p5315PhosPhos20Phos_CBP,p5315PhosPhos20Phos37Phos_C- BP);
t_p5320Phos = SUM(t_p5320Phos_NoCBP,t_p5320Phos_CBP); t_p5337Phos =
SUM( p5337Phos,p5320Phos37Phos,p5320Phos37Phos_CBP,
p5315Phos37Phos,p5315Phos20Phos37Phos, p5315Phos20Phos37Phos_CBP,
p5315PhosPhos37Phos,p5315PhosPhos20Phos- 37Phos, p5315PhosPhos20Pho
s37Phos_CBP); a_p53 = SUM(p5315PhosPhos20Phos37Phos_CBP);
t_p53_NoMdm2 = SUM( p53,p5320Phos,p5320Phos_CBP,
p5315Phos,p5315Phos20Phos,p5315Phos20- Phos_CBP,
p5315PhosPhos,p5315PhosPhos20Phos,p5315PhosPhos20Phos_CBP- ,
p5337Phos,p5320Phos37Phos,p5320Phos37Phos_CBP,
p5315Phos37Phos,p5315Phos20Phos37Phos, p5315Phos20Phos37Phos_CBP,
p5315PhosPhos37Phos,p5315PhosPhos20Phos37Phos, p5315PhosPhos20Pho
s37Phos_CBP); t_p53_Mdm2 = SUM(Mdm2_p5337Phos,Mdm2_p5337Phos); };
save_timeseries = { variables = (t_p5315Phos, t_p5315PhosPhos,
t_p5320Phos, t_p5337Phos, a_p53, t_p53_NoMdm2); save to directory =
"data"; }; plot = timeseries { plot_title = "Plot1"; simulation_TS
= (t_p5315Phos, t_p5315PhosPhos, t_p5320Phos, t_p5337Phos, a_p53,
t_p53_NoMdm2); }; plot = timeseries { plot_title = "Plot2";
simulation_TS = (Mdm2, ARF_Mdm2, Mdm2_p53, Mdm2_p5337Phos); }; plot
= timeseries { plot_title = "Plot3"; simulation_TS = (p53);
experimental_TS = (p53); };
[0442] This file defines some new chemicals which are used only for
output. They are total levels of certain chemicals, so they are
defined as a mathematical function (SUM) of these other chemicals.
Any chemical can be saved to a file or plotted.
[0443] When the code runs, it produces the following output (the
amount of output can be controlled by specifying the
verbosity):
14 Starting with the following parameter values: ks_p53 .about.
(500, , ) kd_p53 .about. (0.5, , ) kt_p53_Nucl .about. (0.2, , )
kt_p53_Cyto .about. (0.2, , ) There are 4 optimizable parameters
Director has 1 networks Network 1 has 78 states and 76 parameters
Using parameter identifier: Simulated Annealing Executing Serial
Simulated Annealing Initializing temperature ... Done - Initial
cost = 521945 Initial temperature = 10000 New lowest cost = 217203
ks_p53 .about. (1473.71, , ) kd_p53 .about. (3.01649, , )
kt_p53_Nucl .about. (1.10296, , ) kt_p53_Cyto .about. (0.148332, ,
)
[0444] Each time the code finds a new lowest cost, it writes the
chemicals and parameters to disk files in a format which can be
read back in to restart the run. Depending on the verbosity
setting, it also writes the values of the optimizable parameters to
the screen. All optimizable parameters have three values: the
current value, the lower bound and the upper bound. In this
example, the bounds are not specified, so they have their default
values which are determined by taking the original value and
dividing by 10 for the lower bound and multiplying by 10 for the
upper bound. When the code has finished running the optimizer, it
writes out a summary:
15 New lowest cost = 134.676 ks_p53 .about. (148.928, , ) kd_p53
.about. (1.07345, , ) kt_p53_Nucl .about. (1.72851, , ) kt_p53_Cyto
.about. (1.6086, , ) Simulated Annealing Minimizer Starting cost =
521945 Minimum cost = 134.676 Initial temperature = 10000 Final
temperature = 646.108 Number steps generated = 286 Number steps
accepted = 50 Simulated Annealing took 19.489 seconds.
[0445] The code then takes the values found by the simulated
annealing minimizer as starting values for the Levenberg-Marquardt
minimizer. After this minimizer has finished, the code again writes
a summary:
16 New lowest cost = 0.255947 ks_p53 .about. (148.92, , ) kd_p53
.about. (0.285072, , ) kt_p53_Nucl .about. (0.537043, , )
kt_p53_Cyto .about. (2.45473, , ) Reached maximum number of
iterations, solution may not be local minimum Levenberg-Marquardt
Minimizer Number iterations = 5 Number objective function
evaluations = 28 Number Jacobian evaluations = 5 Starting cost =
134.676 Minimum cost = 0.255947 Gradient norm = 38.853 Levenberg
Marquardt took 3.815 seconds.
[0446] The results of the optimization are written back to a set of
output files that mimic the input files, except for the fact that
the original values of the optimizable parameters are replaced with
the new optimized values. The files can then used as input to a new
optimization or simulation run.
[0447] Exemplary Method for the Network Inference Methodology
[0448] Described herein is a system for inferring one or a
population of biochemical interaction networks, including topology
and chemical reaction rates and parameters, from dynamical or
statical experimental data, with or without spatial localization
information, and a database of possible interactions. Accordingly,
the invention, as described herein, provides systems and methods
that will infer the biochemical interaction networks that exist in
a cell. To this end, the systems and methods described herein
generate a plurality of possible candidate networks and then apply
to these networks a forward simulation process to infer a network.
Inferred networks may be analyzed via data fitting and other
fitting criteria, to determine the likelihood that the network is
correct. In this way, new and more complete models of cellular
dynamics may be created.
[0449] FIG. 46 depicts a model generator 12 that creates new model
networks, drawing from a combination of sources including a
population of existing networks 14 and a probable links database
16. Once generated, a parameter-fitting module 18 evaluates the
model network, determining parameter values for the model network
based on experimental data 20. A simulation process 26 may aid the
optimization of the parameters in the parameter-fitting module 18.
An experimental noise module 22 may also be used in conjunction the
parameter-fitting module 18 to evaluate the model's sensitivity to
fluctuations in the experimental data 20. Finally, a cost
evaluation module 24 may test the reliability of the model and
parameters by examining global and local fitness criteria.
[0450] A population of existing networks 14 stores previously
inferred network models in a computer database and may provide
network models to a model generator 12 for the generation of new
network models. Completed network models are added to the
population of existing models 14 for storage, transferred from a
cost-evaluation module 24.
[0451] A probable links database 16 stores data representative of
biochemical interactions obtained from bioinformatics predictions,
and may also include hypothetical interactions for which there is
some support in the published literature. The probable links
database 16 couples with the model generator 12 to provide links
for the formation of new network models where necessary.
[0452] The model generator 12 uses any of a number of model-fitting
techniques that are known to those of skill in the art to generate
new biochemical network models. In one embodiment, the model
generator 12 employs genetic algorithms to generate new networks,
using two networks present in the population. Such genetic
algorithms may use other information to guide the recombination of
networks used in constructing new networks, such as sensitivity
analysis of the parameters of one or both of the parent networks.
They may also use the results of clustering analyses to group
together networks in the population that behave in similar ways
dynamically, and selectively recombine networks belonging to the
same dynamical cluster or, for heterotic vigor, recombine networks
belonging to different dynamical clusters but which fit the data
approximately equally well. In creating the new network model, the
model generator 12 may draw one or more networks from the
population of existing networks 14 and incorporate any number of
possible interactions from the probable links database 16.
Alternatively, the model generator 12 may rely solely on the
probable links database 16, generating a new network model without
relying on the population of existing networks 14.
[0453] In one practice, the model generator 12 uses multiple
evaluation criteria, e.g. finite state machines, to test generated
networks for compatibility with experimental data, as in Conradi et
al. (C. Conradi, J. Stelling, J. Raisch, IEEE International
Symposium on Intelligent Control (2001) `Structure discrimination
of continuous models for (bio)chemical reaction networks via finite
state machines`, p. 138). The model generator may also use Markov
Chain Monte Carlo methods (W. Gilks, S. Richardson and D.
Spiegelhalter, `Markov Chain Monte Carlo in Practice`, Chapman and
Hall, 1996), or variational methods(M. Jordan, Z. Ghahramani, T.
Jaakkola, L. Saul, `An introduction to variational methods for
graphical models`, in `Learning in Graphical Models` (M. Jordan,
ed.), MIT Press, 1998), or loopy belief propagation (J. Pearl,
`Causality: Models, Reasoning and Inference`, Cambridge Univ.
Press, 2000), for inferring the likelihood of a given network
topology, given the experimental data. Network topologies that are
unlikely, given the experimental data, would be accepted at a lower
rate than those that are likely, as in the Metropolis algorithm for
Monte Carlo simulations. The model generator may also use the
results of clustering large-scale or high-throughput experimental
measurements, such as mRNA expression level measurements, perhaps
combined with bioinformatics predictions such as for genes with
common binding sites for transcription factors, or secondary
structure predictions for proteins that may be possible
transcription factors, to generate models consistent with these
clustering and bioinformatics results, in combination or singly.
The model generator may also include reactions suggested by a
control theory based module, which can evaluate portions of a given
network in the population and modify them according to calculations
based on robust control theory (F. L. Lewis, Applied Optimal
Control and Estimation, (Prentice-Hall, 1992)).
[0454] As will be understood by one of ordinary skill in the art,
the systems and methods described herein allow for generating a
population of networks and evaluating predictions, from this
population in a manner that is similar or equivalent to a Monte
Carlo evaluation, of the likelihood that the model is correct, in
the Bayesian sense over the ensemble of all networks, weighted by
the a priori measure of the space of networks. With model
generation complete, the newly generated network model passes from
the model generator 12 to a parameter fitting module 18 for
optimization of the network parameters.
[0455] A parameter fitting module 18 optimizes the model parameters
received from the model generator 12 using experimental data 20 as
a calibration point, either in a single step or by coupling with a
simulation module 26 for iterative parameter fitting. Optimization
methods may be according to any global or local routine or
combination of routines known to one of skill in the art. Examples
include, but are not limited to local optimization routines such as
Levenberg-Marquardt, modified Levenberg-Marquardt, BFGS-updated
secant methods, sequential quadratic programming, and the
Nelder-Mead method, or global optimization routines such as
simulated annealing or adaptive simulated annealing, basic
Metropolis, genetic algorithms, and direct fitting. Following
parameter optimization, the parameter fitting module 18 passes the
network model to a cost evaluation module 24.
[0456] The experimental data 20 consists of qualitative or
quantitative experimental data, such as mRNA or protein levels,
stored in a computer database. The experimental data 20 may be
obtained through any of a variety of high-throughput data
collection techniques known to one of skill in the art, including
but not limited to immunoprecipitation assays, Western blots or
other assays known to those of skill in the art, and gene
expression from RT-PCR or oligonucleotide and cDNA microarrays. The
experimental data 20 couples directly with the parameter fitting
module 18 for parameter optimization, and possibly with an
experimental noise module 22. In other practices the systems and
methods described herein employ other types of data, including, for
example, spatial localization data. Preferably the model has
(x,y,z,t) spatial and temporal coordinates for components as well.
Confocal microscopy is one of the technologies for getting both
dynamical and spatial localization. One example of why this is
important, is that the total levels of protein A may not change at
all as a result of the perturbation. But its levels in the cytosol
versus nucleus may be changing as a result of the perturbation
whereby A is getting translocated from cytosol to nucleus to
participate in other processes. Our inference may use both
dynamical and static data, as well as information on spatial
localization. An experimental noise module 22 may be used to
provide an indication of the model's sensitivity to small
variations in experimental measurements. The noise module 22 acts
as an interim step between the experimental data 20 and the
parameter fitting module 18, introducing variations into the
experimental data 20 for evaluation following parameter
optimization in a cost-evaluation module 24. The noise generation
could be implemented by modeling the uncertainty in any given
experimental observation by an appropriate distribution (e.g.
log-normal for expression data) and picking noise values as
dictated by the distribution for that experimental observation.
[0457] With a completed biochemical network model, an optional cost
evaluation module 24 may evaluate the network model received from
the parameter fitting module 18 according to cost or fitness
criteria. The cost evaluation module 24 ranks a model's reliability
according to the chosen fitness or cost criteria. The criteria
employed by the cost evaluation module 24 may include, but are not
limited to: (1) insensitivity of the model to changes in the
initial conditions or chemical reaction parameters, (2) robustness
of the model to the random removal or addition of biochemical
interactions in the network, (3) insensitivity to variations in the
experimental data (with variations introduced into the experimental
data in the experimental noise data 22), and (4) overall
bioinformatics costs associated with the model. Examples of
bioinformatics costs are the number of gene prediction algorithms
that simultaneously agree on a particular gene, the number of
secondary structure prediction algorithms that agree on the
structure of a protein, and so on. Coupled to this, some
bioinformatics algorithms allow comparison to synthetically
generated sequence (or other) data, thereby allowing the
calculation of likelihoods or confidence measures in the validity
of a given prediction. The cost evaluation module 24 then adds the
new network model and the results of its cost criteria to the
population of existing networks 14.
[0458] Models in the population of existing networks continue to be
evaluated and tested by adding and removing links in iterative
operations of the system herein described. There is no specific
starting point in the system. Users of the system may generate
networks entirely from the probable links database 16, or from a
combination of the probable links database 16 with the population
of existing networks 14. Iterative refinement may continue until a
single network attains a goodness of fit to experimental data,
perhaps combined with low costs for dynamical robustness or other
criteria, below a user defined threshold, or a stable dynamically
similar cluster of networks emerges from the population of
networks. This stable cluster may then be used to compute robust
predictions by averaging over the predictions of elements of the
cluster of networks, in a cost-weighted average, where the costs
include, but are not limited to, goodness of fit to the
experimental data, dynamical robustness, probabilistic or exact
evaluation of insensitivity to experimental noise and/or parameter
values. Thus networks with lower costs contribute more to
predictions than networks with higher costs. The refinement of the
pool of networks may be continued until the (average or best)
goodness of fit of the networks in the stable cluster is below some
user defined threshold, or until the number of networks in the
cluster is above some user defined threshold. In the case of the
single network that may be the result of the inference process, the
single network may be solely used for generating predictions.
[0459] The depicted process shown in FIG. 1 can be executed on a
conventional data processing platform such as an IBM PC-compatible
computer running the Windows operating systems, or a SUN
workstation running a Unix operating system. Alternatively, the
data processing system can comprise a dedicated processing system
that includes an embedded programmable data processing system. For
example, the data processing system can comprise a single board
computer system that has been integrated into a system for
performing micro-array analysis. The process depicted in FIG. 46
can be realized as a software component operating on a conventional
data processing system such as a Unix workstation. In that
embodiment, the process can be implemented as a C language computer
program, or a computer program written in any high level language
including C++, Fortran, Java or basic. The process may also be
executed on commonly available clusters of processors, such as
Western Scientific Linux clusters, which are able to allow parallel
execution of all or some of the steps in the depicted process.
[0460] Accordingly, the systems and methods described herein
include systems that create a pool of candidate or possible
networks that have been generated to match data, including data
that is biologically realistic as it arises from relevant
literature or experiments. The systems described herein may, in
certain embodiments, apply a discriminator process to the generated
pool of possible networks. In an iterative process, the system may
employ pools identified by the discriminator process as data that
may be applied to a network generation module. The network
generation module can process these possible networks with data
from the probable links database to generate output data that can
be processed by the fitting module as described above. In this way
the systems and methods described herein may derive predictions
from a pool of networks, instead of processing biological data to
generate a single unique network.
[0461] Those skilled in the art will know or be able to ascertain
using no more than routine experimentation, many equivalents to the
embodiments and practices described herein. Accordingly, it will be
understood that the invention is not to be limited to the
embodiments disclosed herein, but is to be interpreted as broadly
as allowed under the law.
[0462] This methodology has been practiced on synthetic networks of
25, 50, and 100 compoents. By a synthetic network we mean one that
is componsed of many interacting components, but those components
may not necessarily have a basis in the literature. These were
derived as a test case for the methodology. FIG. 47 dispalys an
example of a 100 node network where 50% of the interactions are
removed to reflect what one normally deals with in regards to real
biological networks--most of the interactions are not known. We
then used the methodology to reconstrict the original links or
infer back the original networks. The network inference algorithm
gives back populations of networks that satisfy the cost criteria
described above and reproduces the dynamical behavior of the
original network. Predictions are then dispalyed as a cost weighted
average from this population. FIG. 48 displays the cost function to
fitting the data as one changes or perturbs the links for a 25
component network. We note that the starting network has a very
high cost on the order of millions. One link away and the cost is
drastically reduced indicating the need to infer missing components
and not just constrain parameter values via parameter optimization.
FIGS. 49-50 contain the predicted time course data from chemicals
for which we had observed data for and unobserved data for (the
curves labeled (1) are for the experimental time course and those
labeled (2) are for the reconstructed time course from the network
inference methodology described above). Not only are we able to
resonstrict the dynamical behavior for the observed chemicals, but
we are also able to predict the trends in the unobserved chemicals
as well.
[0463] Particular Uses of the Methods of the Invention
[0464] The methods in the invention, as described, can be used to
perform dread discovery. In addition to finding specific targets,
series of targets, therapeutic agents and combinations of
therapeutic agents, the methods of the invention can be used to
determine which populations of patients have specific targets and
are therefore amenable to treatment with specific therapeutics,
based upon the biological data representing those populations. In
another embodiment of the invention, the biological data of
specific persons can be used in the simulations of the invention to
find the best therapeutic strategy for treating that person, i.e.
the dose, time, order, etc. of different therapeutics affecting
specific targets in that person.
[0465] In still other important embodiments, the methods of the
invention can contribute to finding useful therapies for "failed
compounds", i.e. compounds which have not performed well in
clinical trials, by altering offering the combinations of targets
for such compounds, combining other therapeutic compounds with the
failed compounds, determining specific populations to treat, etc.
In yet another use of the invention, simulations can be used to
identify a molecular marker as a predictor of a disease condition
such that diagnosing physicians may perform analytical tests for
such markers as a precondition to diagnosing that condition. In yet
another embodiment of the invention, compounds which are found to
have therapeutic value can be altered in structure or function to
make them more effective, e.g. by reducing the number of targets
which are addressed, creating higher binding affinities in the
therapeutic, etc. The biological data can be used to infer what
target the drug is impacting based on network inferences.
* * * * *
References